Counting Prime Graphs and Point-Determining Graphs Using...

Counting Prime Graphs and Point-Determining Graphs

Using Combinatorial Theory of Species

A Dissertation

Presented to

The Faculty of the Graduate School of Arts and Sciences

Brandeis University

Department of Mathematics

Ira Gessel, Advisor

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

by

Ji Li

August, 2007

The signed version of this signature page is on file at the Graduate School of Arts

and Sciences at Brandeis University.

This dissertation, directed and approved by Ji Li’s committee, has been accepted

and approved by the Faculty of Brandeis University in partial fulfillment of the

requirements for the degree of:

DOCTOR OF PHILOSOPHY

Adam Jaffe, Dean of Arts and Sciences

Dissertation Committee:

Ira Gessel, Dept. of Mathematics, Chair.

Susan Parker, Dept. of Mathematics

Alex Postnikov, Dept. of Mathematics, MIT

c© Copyright by

Ji Li

2007

Dedication

To Kailang Li and Youzhi Jiang.

Lu Man Man Qı Xiu Yuan Xi,

Wu Jiang Shang Xia Er Qiu Suo.

by: Qu Yuan

The way is long with obstacles,

I’m questing for the truth to and fro.

iv

Acknowledgments

I would like to thank my advisor, Professor Ira Gessel, for his constant help and

support, without which this work would never have existed.

I am grateful to the members of my dissertation defense committee, Professor

Susan Parker from Brandeis University and Professor Alex Postnikov from MIT.

I also owe thanks to the faculty, to my fellow students, and to the kind and

supportive staff of the Brandeis Mathematics Department.

v

Abstract

Counting Prime Graphs and Point-Determining Graphs Using

Combinatorial Theory of Species

A dissertation presented to the Faculty of theGraduate School of Arts and Sciences of Brandeis

University, Waltham, Massachusetts

by Ji Li

In this thesis, we enumerate different types of graphs in light of the combinatorial

theory of species initiated by Joyal [13]. The prime graphs with respect to the

Cartesian multiplication is enumerated using the exponential composition of species,

which is constructed based on the arithmetic product of species studied by Maia

and Mendez [18]. The point-determining graphs and bi-point-determining graphs are

enumerated by finding functional equations relating them to the species of graphs.

vi

Contents

List of Figures ix

Chapter 1. Groups Actions and Polya’s Cycle Index Polynomial 1

1.1. Symmetric Groups and Group Actions 1

1.2. Polya’s Cycle Index Polynomials 5

1.3. Exponentiation Group 8

Chapter 2. Combinatorial Theory of Species 15

2.1. Definition of Species 15

2.2. Species Operations 21

2.3. Molecular Species 25

2.4. Compositional Inverse of E+ 31

2.5. Multisort Species 35

2.6. Arithmetic Product of Species 38

Chapter 3. Cartesian Product of Graphs and Prime Graphs 48

3.0. Introduction 48

3.1. Cartesian Product of Graphs 49

3.2. Labeled Prime Graphs 54

3.3. Unlabeled Prime Graphs 61

3.4. Exponential Composition with a Molecular Species 67

3.5. Exponential Composition 78

vii

3.6. Cycle Index of Prime Graphs 82

Chapter 4. Point-Determining Graphs 86

4.0. Introduction 86

4.1. Point-Determining Graphs and Co-Point-Determining Graphs 87

4.2. Connected Point-Determining Graphs and Connected Co-Point-

Determining Graphs 91

4.3. Bi-Point-Determining Graphs 96

4.4. 2-Colored Graphs and Connected 2-colored Graphs 112

4.5. Point-Determining 2-Colored Graphs 117

Appendix A. Index of Species 122

Appendix B. General Notations 124

Appendix. Bibliography 127

viii

List of Figures

1.1 Group action of A > B. 3

1.2 Group action of A × B. 3

1.3 Group action of B ≀ A. 4

1.4 Group action of BA. 9

1.5 An element of BA. 10

1.6 Cycle type of an element of BA. 10

2.1 Transport of G -structures. 16

2.2 Species associated to a graph. 17

2.3 A linear order and a permutation. 20

2.4 Addition of species. 21

2.5 Multiplication of species. 23

2.6 Composition of species. 23

2.7 The formula (Xm/A) · (Xn/B) = Xm+n/(A × B). 31

2.8 The formula (Xm/A) (Xn/B) = Xmn/(B ≀ A). 31

2.9 Unlabeled connected graphs. 35

2.10 Multisort species. 37

2.11 The 2-sort species of rooted trees. 38

2.12 A rectangle. 39

ix

2.13 Another rectangle. 39

2.14 A 3-rectangle. 40

2.15 Arithmetic product of species. 42

2.16 Unit of the arithmetic product. 42

2.17 The species E2 ⊡ E2. 43

2.18 The formula (Xm/A) ⊡ (Xn/B) = Xmn/(A × B). 44

2.19 The E2 ⊡ E2-structures. 45

2.20 An element of A × B. 47

3.1 The Cartesian product of graphs. 50

3.2 A prime decomposition. 51

3.3 The automorphism group of a prime power. 56

3.4 The species E2〈E2〉. 69

3.5 The formula ((Xn/B)⊡m)/A = Xnm

/BA. 71

3.6 Unlabeled prime graphs. 84

3.7 Unlabeled non-prime graphs. 85

4.1 Labeled point-determining graphs and co-point-determining graphs. 88

4.2 Transform a graph into a point-determining graph. 89

4.3 Unlabeled point-determining graphs. 91

4.4 Unlabeled connected point-determining graphs and unlabeled connected

co-point-determining graphs. 95

4.5 A phylogenetic tree. 96

4.6 A functional equation for the species of phylogenetic trees. 97

x

4.7 Unlabeled phylogenetic trees. 98

4.8 Unlabeled phylogenetic trees corresponding to the species XE 22 . 99

4.9 An alternating phylogenetic tree. 100

4.10 An illustration of (g, h). 101

4.11 Operations OQ and OP . 103

4.12 An illustration of (g′, h′). 103

4.13 Alternating applications of operations OP and OQ. 105

4.14 Construct a graph from a given triple (π, ϕ, γ). 107

4.15 Unlabeled bi-point-determining graphs. 109

4.16 The 2-sort species of 2-colored graphs. 112

4.17 Unlabeled 2-colored graphs counted by the number of edges. 114

4.18 Unlabeled 2-colored graphs. 115

4.19 Unlabeled connected 2-colored graphs. 116

4.20 The formula Ps(X, Y ) = (1 + X)(1 + Y ) E (Pc≥2(X, Y )). 118

4.21 The formula P(X, Y ) = (1 + X + Y ) E (Pc≥2(X, Y )). 118

4.22 Unlabeled point-determining 2-colored graphs. 120

4.23 Unlabeled point-determining 2-colored graphs with 5 vertices. 121

xi

CHAPTER 1

Groups Actions and Polya’s Cycle Index Polynomial

1.1. Symmetric Groups and Group Actions

The symmetric group of order n, denoted Sn, is the group of permutations of [n] =

1, 2, . . . , n. Let λ = (λ1, λ2, . . . ), where the λi are arranged in weakly decreasing

order, be a partition of n, denoted λ ⊢ n. That is, |λ| =∑

i λi = n. Let σ be

a permutation of [n]. We say λ is the cycle type of σ, denoted by λ = c. t.(σ), if

the λi are the lengths of the cycles in the decomposition of σ into disjoint cycles in

weakly decreasing order. Sometimes we write λ = (1c1(λ), 2c2(λ), . . . ), where ck(λ) is

the number of parts of length k in λ for k ≥ 1.

Example 1.1.1. Let σ =(

14

22

35

41

56

63

)= (3, 5, 6)(1, 4)(2). Then the cycle type of σ is

(3, 2, 1), and c1(σ) = c2(σ) = c3(σ) = 1.

It is well-known that the number of permutations of [n] of cycle type

λ = (1c1 , 2c2, . . . , kck)

is

dλ :=n!

c1! 1c1 c2! 2c2 · · · ck! kck,

in which the denominator

zλ := c1! 1c1 c2! 2c2 · · · ck! kck

1

CHAPTER 1. GROUP ACTIONS

is the number of permutations in Sn that commute with a permutation of cycle type

λ.

Definition 1.1.2. An action of a group A on a set S is a function

ρ : A × S → S,

where for x ∈ A and s ∈ S, ρ(x, s) is written x · s. We say this action is natural if

both of the following conditions are satisfied:

x · (y · s) = (xy) · s, idA ·s = s,

for any x, y ∈ A and s ∈ S.

Let A be a subgroup of Sm, and let B be a subgroup of Sn. We construct new

groups as described below.

Definition 1.1.3. We define two groups A > B and A × B both isomorphic to the

product of A and B, where A > B is a subgroup of Sm+n and A × B is a subgroup

of Smn.

The elements of the product groups are of the form (a, b), where a ∈ A and b ∈ B.

The group operation is defined by

(a1, b1) · (a2, b2) = (a1a2, b1b2),

where a1 and a2 are elements of A, and b1 and b2 are elements of B.

The group A > B acts on the set [m + n] by

(1.1.1) (a, b)(i) =

a(i), if i ∈ 1, 2, . . . , m,

b(i − m) + m, if i ∈ m + 1, m + 2, . . . , m + n.

2


Figure 1.1 illustrates the action of an element (a, b) of the group A > B.

b

a

Figure 1.1. Group action of A > B.

The group A × B acts on the set [m] × [n] and may therefore be viewed as a

subgroup of Smn. The action of an element of A × B on an element of [m] × [n] is

given by

(a, b)(i, j) = (a(i), b(j)),

for all i ∈ [m] and j ∈ [n].

Figure 1.2 illustrates the action of an element (a, b) of the group A × B.

b

a

Figure 1.2. Group action of A × B.

Definition 1.1.4. The wreath product of A and B, denoted B ≀ A is the group in

which the elements are ordered pairs (α, τ), where α is a permutation in A and τ is

3


a function from [m] to B. An element (α, τ) of B ≀ A acts on the set [m] × [n] by

(α, τ)(i, j) = (αi, τ(i)j),

for all i ∈ [m] and j ∈ [n].

Figure 1.3 illustrates the action of an element (α, τ) of the group B ≀ A.

α

τ(1)

τ(m)

τ(2)

Figure 1.3. Group action of B ≀ A.

The composition of two elements (α, τ) and (β, η) of B ≀ A is given by

(α, τ)(β, η) = (αβ, (τ β)η),

where β ∈ A is viewed as a function from [m] to [m], and (τ β)η denotes the

point-wise multiplication of τ β and η, both functions from [m] to B.

Again B ≀ A can be identified with a subgroup of Smn. Note that the order of

B ≀ A is∣∣B ≀ A

∣∣ = |A| · |B|m.

Example 1.1.5. Let A be the symmetric group of order 2, and let B be the cyclic

group of order 3. The element (α, τ) in B ≀ A, where α = (1, 2), τ(1) = id and

τ(2) = (1, 2, 3), acts on the set [2] × [3] by

(1, 1) 7→ (2, 1) (2, 1) 7→ (1, 2)

(1, 2) 7→ (2, 2) (2, 2) 7→ (1, 3)

(1, 3) 7→ (2, 3) (2, 3) 7→ (1, 1)

4


Therefore, the element (α, τ) of B ≀ A is a 6-cycle.

The following Theorem is a generalization of the Cauchy-Frobenius Theorem,

alias Burnside’s Lemma. For the proof of a more general result, with applications

and further references, see Robinson [25]. Another application is given in [6].

Theorem 1.1.6. (Cauchy-Frobenius) Suppose that a finite group M ×N acts on

a set S. The groups M and N , considered as subgroups of M ×N , also act on S. The

group N acts on the set of M-orbits. Then for any g ∈ N , the number of M-orbits

fixed by g is given by

1

|M |

∑

f∈M

fix(f, g),

where fix(f, g) denotes the number of elements in S that are fixed by (f, g) ∈ M ×N .

1.2. Polya’s Cycle Index Polynomials

Definition 1.2.1. Let λ be a partition of n. The power sum symmetric function in

the variables x1, x2, . . . [27, p. 297] indexed by λ, denoted pλ, is defined by

pn = pn[x] =∑

i

xni , n ≥ 1

pλ = pλ[x] = pλ1pλ2 · · · =∏

k≥1

pck(λ)k , if λ = (λ1, λ2, . . . ) = (1c1(λ), 2c2(λ), . . . ).

The pλ form a basis for the ring of symmetric functions in the variables x1, x2, . . . .

We can also define power sum symmetric functions in the variables y1, y2, y3, . . . ,

written as pλ[y], in a similar fashion.

Definition 1.2.2. The operation called plethysm on the ring of symmetric functions

(see Stanley [27, p. 447]) is defined to be such that if f is a symmetric function in

the variables x1, x2, . . . , then the plethysm f pn is obtained from the expression for

5


f in terms of the xi’s by replacing each xi with xni . More generally, if f and g are

arbitrary symmetric functions, where g is expressed as a function of the power sum

symmetric functions, i.e., g = g(p1, p2, . . . ), then f g is the polynomial obtained

from f by replacing each variable pk by g(pk, p2k, . . . ).

It follows immediately that pm pn = pmn for all positive integers m, n.

Next, we introduce an operation ⊠ on power sum symmetric functions that was

studied by Harary [10], and by Maia and Mendez [18].

Definition 1.2.3. We define an operation ⊠ on the symmetric functions by letting

pν := pλ ⊠ pµ,

where

ck(ν) =∑

lcm(i,j)=k

gcd(i, j) ci(λ)cj(µ),

in which lcm(i, j) denotes the least common multiple of i and j, and gcd(i, j) de-

notes the greatest common divisor of i and j, and then extending it to all symmetric

functions through bilinearity.

Proposition 1.2.4. The operation ⊠ as defined by Definition 1.2.3 is commuta-

tive and associative. That is, for any partitions λ, µ and ν,

(commutativity) pλ ⊠ pµ = pµ ⊠ pλ,

(associativity) (pλ ⊠ pµ) ⊠ pν = pλ ⊠ (pµ ⊠ pν).

Proof. The commutativity is clear.

For the associativity, we write

(pλ ⊠ pµ) ⊠ pν = pα,

6


pλ ⊠ (pµ ⊠ pν) = pβ.

Using the fact that for any positive integers i and j,

ij = lcm(i, j) gcd(i, j),

we calculate the number of cycles of length l, for any integer l, in the partition α as

follows:

cl(α) =∑

lcm(i,j,k)=l

gcd(i, j) gcd(lcm(i, j), k) ci(λ)cj(µ)ck(ν)

=∑

lcm(i,j,k)=l

ij

lcm(i, j)

lcm(i, j) k

lcm(lcm(i, j), k)ci(λ)cj(µ)ck(ν)

=∑

lcm(i,j,k)=l

ijk

lcm(i, j, k)ci(λ)cj(µ)ck(ν)

=∑

lcm(i,j,k)=l

ijk

lci(λ)cj(µ)ck(ν).

The calculation for the partition β is similar.

The properties of ⊠ allow us to talk about the ⊠ operation on a set of partitions.

Let λ(i), i = 1, 2, . . . , l, be partitions of n1, n2, . . . , nl, respectively. We have

pλ(1) ⊠ pλ(2) ⊠ · · · ⊠ pλ(l) = pν ,

where ν is a partition of N = n1n2 · · ·nl. It is easy to verify that

ck(ν) =∑

lcm(j1,j2,...,jl)=k

∏lr=1 jr

k

l∏

i=1

cji

(λ(i)

).

7


Definition 1.2.5. Let σ ∈ Sn. The cycle type monomial of σ, denoted Z(σ), is the

monomial in the variables p1, p2, . . . , pn defined by

Z(σ) =

n∏

k=1

pck(σ)k = pc.t.(σ).

In other words, the cycle type monomial of σ is the power sum symmetric function

index by the cycle type of σ.

Let A be a subgroup of Sn. The cycle index polynomial of A, defined by Polya [22,

pp. 64–65], is

Z(A) = Z(A; p1, p2, . . . , pn) =1

|A|

∑

σ∈A

Z(σ).

Polya [21] (translated in [22]) gave the cycle index polynomials of the product

and the wreath product of two permutation groups.

Theorem 1.2.6. (Polya) Let A and B be permutation groups on [m] and [n],

respectively. Let A > B be the product of A and B acting on [m + n], and let B ≀A be

the wreath product of A and B acting on [mn]. Then the cycle index polynomials of

A > B and B ≀ A are

Z(A > B) = Z(A)Z(B),

Z(B ≀ A) = Z(A) Z(B),

where Z(A) Z(B) denotes the plethysm of Z(A) with Z(B).

1.3. Exponentiation Group

Let A be a subgroup of Sm, and let B be a subgroup of Sn. We introduce in the

following another representation of the wreath product (Definition 1.1.4).

8


Definition 1.3.1. The exponentiation group BA, studied by Palmer [19], is isomor-

phic to the wreath product B ≀A. That means, the set of elements of BA is the same

as the set of elements of B ≀A, and hence the order of BA is the same as the order of

B ≀ A, i.e.,

|BA| = |A| · |B|m.

However the exponentiation group BA acts on [n]m, identified with the set of functions

from [m] to [n], instead of [m] × [n]. Given any α ∈ A and τ ∈ Bm, the element

(α, τ) ∈ BA sends each element f ∈ [n]m to the element g ∈ [n]m defined by

((α, τ)f)(i) = g(i) = τ(i)(f(α−1i)),

for any i ∈ [m]. Therefor, BA can be identified with a subgroup of Snm.

Figure 1.4 illustrates the action of an element (α, τ) of the group BA.

τ(1)

τ(5

) τ(2)

τ(3)τ(4)

α

α

αα

α

Figure 1.4. Group action of BA.

Example 1.3.2. Let A, B, α and τ be the same as in Example 1.1.5.

9


Let f : [2] → [3] be such that f(1) = 1 and f(2) = 3. We see that the image of f

under the action of (α, τ) ∈ BA is the function g : [2] → [3] such that g(1) = 3 and

g(2) = 2, as shown in Figure 1.5.

f(2) = 3

f g(α, τ)

g(1) = 3

g(2) = 2

f(1) = 1

Figure 1.5. The element (α, τ) of BA sends f to g.

The action of (α, τ) on the set of functions from [2] to [3] is illustrated in Figure 1.6.

Hence the cycle type of (α, τ) is (6, 3).

Figure 1.6. The cycle type of (α, τ) is (6, 3).

The cycle index polynomial of the exponentiation group was given by Palmer and

Robinson [20]. They defined the following operators Ik for positive integers k.

Let R = Q [p1, p2, . . . ] be the ring of polynomials with the operation ⊠ as defined

by Definition 1.2.3. Palmer and Robinson defined for positive integers k the Q-linear

operators Ik on R as follows:

10


Let λ = (λ1, λ2, . . . ) be a partition of n. The action of Ik on the monomial pλ is

given by

(1.3.1) Ik(pλ) = pγ ,

where γ = (γ1, γ2, . . . ) is the partition of nk with

cj(γ) =1

j

∑

l|j

µ

(j

l

)( ∑

i | l/ gcd(k,l)

ici(λ)

)gcd(k,l)

.

Furthermore, Ik generates a Q-algebra Ω of Q-linear operators on R. For any

elements I, J ∈ Ω, any r ∈ R and a ∈ Q, we set

(aI)(r) = a(I(r)),

(I + J)(r) = I(r) + J(r),

(IJ)(r) = I(r) ⊠ J(r).(1.3.2)

Remark 1.3.3. As discussed in Palmer and Robinson’s proof of Theorem 1.3.6, if

Im(pµ) = pν , then ν is the cycle type of an element (α, τ) of the exponentiation group

BA acting on [n]m, where α is a permutation in A with a single m-cycle, and τ ∈ Bm

is such that

c. t. (τ(m)τ(m − 1) · · · τ(2)τ(1)) = µ.

Example 1.3.4. We take the same A, B, α and τ from Examples 1.1.5. That is,

A = S2, B = C3, α = (1, 2) and τ(1) = id, τ(2) = (1, 2, 3). Then the cycle type of α

is λ = (2), and the cycle type of τ(2)τ(1) is µ = (3). The cycle type of (α, τ) acting

on the set [3]2 is ν = (6, 3), as shown in Figure 1.6, hence

I2(p3) = p3p6.

11


Definition 1.3.5. Let f1 and f2 be elements of the ring R = Q [p1, p2, . . . ]. We

define the exponential composition of f1 and f2, denoted f1 ∗ f2, to be the image of

f2 under the operator obtained by substituting the operator Ir for the variables pr in

f1.

Note that the operation ∗ is linear in the left parameters, but not on the right

parameters. We call this the partial linearity of the operation ∗.

Let A be a subgroup of Sm, and let B be a subgroup of Sn. Palmer and Robinson

proved the following formula [20, pp. 128–131] for computing the cycle index poly-

nomial of the exponentiation group BA in terms of the cycle index polynomials of A

and B.

Theorem 1.3.6. (Palmer and Robinson) Let A and B be as described in above.

Then the cycle index polynomial of BA is the exponential composition of Z(A) with

Z(B). That is,

(1.3.3) Z(BA) = Z(A) ∗ Z(B).

Example 1.3.7. Continuing Example 1.3.4, we apply Theorem 1.3.6 to compute

the cycle index of the group BA, where A = S2 and B = C3. Note that the order of

BA is∣∣BA

∣∣ = |A| · |B|2 = 18.

We get the cycle index polynomial of BA

Z(BA) = Z(A) ∗ Z(B)

=

[1

2(p2

1 + p2)

]∗

[1

3(p3

1 + 2p3)

]

12


=1

2I1

[1

3(p3

1 + 2p3)

]⊠ I1

[1

3(p3

1 + 2p3)

]+

1

2I2

[1

3(p3

1 + 2p3)

]

=1

2

[1

3I1(p

31) +

2

3I1(p3)

]⊠

[1

3I1(p3

1) +2

3I1(p3)

]+

1

6I2(p3

1) +1

3I2(p3)

=1

18I1(p3

1) ⊠ I1(p31) +

2

9I1(p3) ⊠ I1(p3

1) +2

9I1(p3) ⊠ I1(p3) +

1

6I2(p3

1) +1

3I2(p3)

=1

18p9

1 +2

9p3

3 +2

9p3

3 +1

6p3

1p32 +

1

3p3p6

=1

18(p9

1 + 8p33 + 3p3p

32 + 6p3p6).

The cycle index polynomial Z(BA) tells the structure of the group BA. For

example, the coefficient 6 for the term p3p6 tells that there are 6 elements in BA with

cycle type (6, 3). These elements are precisely those (α, τ), where α is the 2-cycle in

group A, and τ : [2] → B is such that τ(1)τ(2) is a 3-cycle in group B. Moreover, we

observe that the term p33 in Z(BA) with coefficient 8 is contributed in two ways. One

is from those (α, τ) such that α is the identity element of A, and τ(1) and τ(2) are

3-cycles in B, each having two choices; the other is from those (α, τ) such that α is

the identity element of A, one of τ(1) and τ(2) is the identity of B, and the other is

a 3-cycle in B, which gives rise to 4 possibilities.

Remark 1.3.8. Let A and B be the same as in Theorem 1.3.6. Let (α, τ) be an

element of BA, where

c. t.(α) = λ = (λ1, λ2, . . . ),

and τ ∈ Bm is such that the cycle type of τ restricted on each λi-cycle of α is µ(i),

i.e., if the i-th cycle of α is written as

(a1, a2, . . . , aλi),

13


then µ(i) is the cycle type of the permutation

τ(a1)τ(a2) · · · τ(aλi).

Then the cycle type of (α, τ), as an element of the group BA acting on the set of

functions from [m] to [n], is ν for which

pν = Iλ1(pµ(1)) ⊠ Iλ2(pµ(2)) ⊠ · · · .

In such cases, for simplicity and without ambiguity, we write

pν = I(λ; µ(1), µ(2), . . . ).

14

CHAPTER 2

Combinatorial Theory of Species

2.1. Definition of Species

The combinatorial theory of species was initiated by Joyal [13, 12]. In short,

species are classes of “labeled structures”. A formal definition (see Bergeron, Labelle,

and Leroux [2, pp. 1–11]) is given in the following.

Definition 2.1.1. Let B be the category of finite sets with bijections. A species (of

structures) is a functor from B to itself, i.e.,

F : B → B.

Given a species F , we obtain for each finite set U a finite set F [U ], which is called

the set of F -structures on U , and for each bijection σ : U → V a bijection

F [σ] : F [U ] → F [V ],

which is called the transport of F -structures along σ.

We denote by F [n] = F [1, 2, . . . , n] the set of F -structures on [n]. The symmet-

ric group Sn acts on the set F [n] by transport of structures. The Sn-orbits under

this action are called unlabeled F -structures of order n.

Example 2.1.2. The species of graphs is denoted by G . Note that by graphs we

mean simple graphs, that is, graphs without loops or multiple edges. Thus defined,

15

CHAPTER 2. COMBINATORIAL THEORY OF SPECIES

we mean by G [U ] the set of graphs with vertex set U , and by a G -structure on U a

graph with vertex set U . More formally, the species of structures G generates

• for any finite set U , a set of graphs with vertex set U ;

• for any bijection σ : U → V , a bijection G [σ] : G [U ] → G [V ].

1 53

42

a ec

db

U = 1, 2, 3, 4, 5

V = a, b, c, d, e

σ

Figure 2.1. A bijection σ from U to V induces a bijection G [σ], thetransport of G -structures along σ, sending a graph with vertex set Uto a graph with vertex set V .

Example 2.1.3. We give a list of examples of species.

• The empty species 0 is defined by setting 0[U ] = ∅ for all U .

• The singleton set species 1 is defined by setting

1[U ] =

U, if U = ∅,

∅, otherwise.

• The species of singletons X is defined by setting

X[U ] =

U, if |U | = 1,

∅, otherwise.

• The species of sets E is defined by setting E [U ] = U for each finite set U .

16


• The species of linear orders L . In particular, the species of linear orders on

n-element sets is denoted by Xn.

• The species of connected graphs G c.

• The species of complete graphs K .

• The species of permutations Φ.

Example 2.1.4. For any graph G, we define the species associated to G, denoted

OG, to be such that

a) for each finite set U , OG[U ] is the set of graphs isomorphic to G with vertex

set U . Note that L(G) = |OG[n]| .

b) for each bijection σ : U → V , where |U | = |V |, OG[σ] is a bijection from

OG[U ] to OG[V ], sending a graph H isomorphic to G with vertex set U to a

graph OG[σ](H) whose vertex labeling is obtained from the vertex labeling

of H by replacing each label u ∈ U with σ(u).

It is straightforward to see that the bijections OG[σ] further satisfy

i. for all bijections σ : U → V and τ : V → W , OG[τσ] = OG[τ ]OG[σ].

ii. if U = V and σ is the identity map on U , then OG[σ] is also the identity map

on OG[U ].

G OG[a, b, c, d, e]

e

a

b

b e

dc

d a

b c

d

a c

ba

ed

c

b

ce

a

e

d

Figure 2.2. The 5 OG-structures on the vertex set a, b, c, d, e fora given graph G.

17


Definition 2.1.5. Each species F is associated with three generating series. First,

the exponential generating series of the species F , given by

F (x) =∑

n≥0

|F [n]|xn

n!,

counts labeled F -structures.

Second, the type generating series of the species F , given by

F (x) =∑

n≥0

fn xn,

where fn is the number of unlabeled F -structures of order n, counts unlabeled F -

structures.

The last, but also the most important, associated series is called the cycle index

of the species F :

ZF = ZF (p1, p2, . . . ) =∑

n≥0

(∑

λ⊢n

fix F [λ]pλ

zλ

).

In this definition, fix F [λ] denotes the number of F -structures on [n] fixed by F [σ],

where σ is a permutation of [n] with cycle type λ, and pλ is the power sum symmetric

function indexed by the partitions λ of n.

The following theorem (see [2]) illustrates the importance of the cycle index in

the theory of species.

Theorem 2.1.6. (Bergeron, Labelle and Leroux) For any species of structures F ,

we have

F (x) = ZF (x, 0, 0, . . . ),

F (x) = ZF (x, x2, x3, . . . ).

18


Remark 2.1.7. Consider the functorial aspects of the species of sets E and the

species of complete graphs K . We see that there is a natural transformation α that

produces for every finite set U a bijection between E [U ] and K [U ], namely, sending

the set U to the complete graph with vertex set U . Furthermore, the following

diagram commutes for any finite sets U , V and any bijection σ : U → V :

E [U ]E [σ]

−−−→ E [V ]

α

yyα

K [U ]K [σ]−−−→ K [V ]

In this case we call these two species isomorphic to each other, denoted E = K .

The general definition of two species being isomorphic to each other is similar. As

pointed out on [2, p. 21], the concept of isomorphism is compatible with the transition

to series. Therefore, we do not make distinctions between isomorphic species during

calculations.

Example 2.1.8. For Φ and L the species of permutations and the species of linear

orders, we notice that their exponential generating series are identical:

Φ(x) = L (x) =1

1 − x,

while their type generating series and cycle indices differ:

Φ(x) =∏

k≥1

1

1 − xk, L (x) =

1

1 − x,

ZΦ =∏

k≥1

1

1 − pk, ZL =

1

1 − p1.

This is because that although there are same number of permutations and linear

orders on a given finite set U , the permutations and linear orders are not transported

19


in the same manner along bijections. In fact, a linear order admits only a single

automorphism, while a permutation admits many automorphisms in general.

2 1

342 5

3 5 4 1

a) b)

Figure 2.3. a) A linear order on [5] admits only a single automor-phism. b) A permutation on [5] admits six automorphisms.

Example 2.1.9. Let G be the species of graphs. It is well-known that the exponen-

tial generating series of G is given by

G (x) =∑

n≥0

2(n2)

xn

n!.

The cycle index of G was given on [2, p. 76]:

(2.1.1) ZG =∑

n≥0

(∑

λ⊢n

fix G [λ]pλ

zλ

),

where

fix G [λ] = 212

∑i,j≥1 gcd(i, j) ci(λ)cj(λ)− 1

2

∑k≥1(k mod 2) ck(λ),

in which ci(λ) denotes the number of parts of length i in λ.

The above formulas permit us to write out the first several terms of the associated

series of G using Maple:

G (x) = 1 +x

1!+ 2

x2

2!+ 8

x3

3!+ 64

x4

4!+ 1024

x5

5!+ 32768

x6

6!+ 2097152

x7

7!

+ 68435456x8

8!+ 68719476736

x9

9!+ · · · ,

G (x) = 1 + x + 2x2 + 4x3 + 11x4 + 34x5 + 156x6 + 1044x7 + 12346x8 + · · · ,

20


ZG = 1 + p1 + (p21 + p2) +

(4

3p3

1 + 2p1p2 +2

3p3

)

+

(8

3p4

1 + 4p21p2 + 2p2

2 +4

3p1p3 + p4

)

+

(128

15p5

1 +32

3p3

1p2 + 8p1p22 +

8

3p2

1p3 +4

3p2p3 + 2p1p4 +

4

5p5

)+ · · · ,

2.2. Species Operations

We describe in this section some frequently used operations on species, namely,

the sum, product, and substitution of species. Readers are referred to [2, pp. 1–58]

for more detailed definitions of F1 + F2, F1F2, F1(F2), F1 × F2, F •1 , F ′

1 for arbitrary

species F1 and F2.

Definition 2.2.1. The sum of F1 and F2 is denoted by F1+F2. An F1+F2-structure

on a finite set U is either an F1-structure on U or an F2-structure on U .

or

F1 F2F1 + F2

=

Figure 2.4. Addition of species F1 + F2.

The formulae for F1 + F2 are given by

(F1 + F2)(x) = F1(x) + F2(x),

(F1 + F2)(x) = F1(x) + F2(x),

ZF1+F2 = ZF1 + ZF2.

21


Each species F gives rise to a canonical decomposition

F = F0 + F1 + F2 + · · · + Fn + · · · ,

where Fnn≥0 is the family of species defined by setting, for each n,

Fn[U ] =

F [U ], if |U | = n,

∅, otherwise.

If F = Fn, then we say the species F is concentrated on the cardinality n. We denote

by F+ the species of nonempty F -structures:

F+ = F1 + F2 + · · · + Fn + · · · ,

F+[U ] =

F [U ], if |U | ≥ 1,

∅, if |U | = 0.

Example 2.2.2. The species of sets E and the species of linear orders L can be

decomposed as

E = E0 + E1 + E2 + · · · = 1 + X + E2 + · · · .

L = L0 + L1 + L2 + · · · = 1 + X + X2 + · · · .

Definition 2.2.3. The product of F1 and F2 is denoted by F1 ·F2. An F1F2-structure

on a finite set U is of the form (π; f1, f2), where π is an ordered partition of U with

two blocks U1 and U2, Ui possibly empty, and fi is an Fi-structure on Ui for each i.

The formulae for the associated series of F1F2 are given by

(F1F2)(x) = F1(x)F2(x),

(F1F2)(x) = F1(x)F2(x),

22


ZF1F2 = ZF1ZF2.

F1 · F2

F1 F2

=

Figure 2.5. Multiplication of species F1 · F2.

Definition 2.2.4. The composition of F1 and F2 is denoted by F1 F2, or equiva-

lently, F1(F2). An F1(F2)-structure on a finite set U is a tuple of the form (π, f, γ),

where π is a partition of U , f is an F1-structure on π, and γ is a set of F2-structures

on the blocks of π.

==

F1F1 F2

F2

F2

F2

F2

F2

F2F1

Figure 2.6. Composition of species F1 F2.

The formulae for the associated series of F1(F2) are given by

(F1(F2))(x) = F1(F2(x)),

(F1(F2))(x) = ZF1(F2(x), F2(x2), . . . ),

23


ZF1(F2) = ZF1 ZF2,

where is the operation of plethysm on symmetric functions defined by Defini-

tion 1.2.2.

Definition 2.2.5. A group A is said to act naturally on a species F if, for any finite

set U , there is an A-action

ρU : A × F [U ] → F [U ]

so that for each bijection σ : U → V , the following diagram commutes:

A × F [U ]ρU−−−→ F [U ]

idA ×F [σ]

yyF [σ]

A × F [V ]ρV−−−→ F [V ]

Definition 2.2.6. The quotient species (see Bergeron, Labelle, and Leroux [2, p. 159])

of F by A, denoted F/A, is defined based on a group A acting naturally on a species

F such that

a) for each finite set U , the set of F/A-structures on U is the set of A-orbits of

F -structures on U , i.e.,

(F/A)[U ] = F [U ]/A,

b) for each bijection σ : U → V , the transport of structures

(F/A)[σ] : F [U ]/A → F [V ]/A

is induced from the bijection F [σ] that sends each orbit of the action of A on F [U ]

to an orbit of the action of A on F [V ].

24


The bijections (F/A)[σ] satisfy the functorial properties

(F/A)[τσ] = (F/A)[τ ](F/A)[σ], (F/A)[idU ] = id(F/A)[U ],

because of the functoriality of the species F .

The notion of quotient species appeared in [8] and [3] as an important tool in

combinatorial enumeration.

Example 2.2.7. Let E+ be the species of nonempty sets. Then an (E+ ·E+)-structure

on a finite set U is an ordered pair of nonempty subsets (U1, U2) whose union equals

U . The symmetric group S2 acts naturally on the subscripts of Ui for i = 1, 2, with

orbits unordered pairs of such (U1, U2). Thus we get a quotient species (E+ · E+)/S2,

the species of partitions into two blocks, denoted Ψ(2).

Proposition 2.2.8. Let F1 and F2 be two species. Let A be a group acting natu-

rally on both F1 and F2. Then

(2.2.1) (F1 + F2)/A = F1/A + F2/A.

Proof. An F1 + F2-structure on a finite set U is either an F1-structure on U or

an F2-structure on U . It follows immediately that an orbit of A acting on the set of

F1 + F2-structures on U is either an orbit of A acting on the set F1[U ] or an orbit of

A acting on the set F2[U ].

2.3. Molecular Species

Definition 2.3.1. A species of structures M is said to be molecular [31, 30] if there

is only one isomorphism class of M-structures, i.e., if two arbitrary M-structures are

isomorphic.

25


In other words, the species M is molecular if and only if M 6= 0, M is concentrated

on n, and any element in the set M [n] can be sent through some transport of structures

to any other element in M [n].

We introduce in the following a construction of, for any subgroup A of Sn, a

molecular species Xn/A (see Bergeron, Labelle, and Leroux [2, p. 144]). These species

satisfy that Xn/A = Xn/B if and only if A and B are conjugates in Sn. In fact, if M

is any molecular species concentrated on the cardinality n, and s is any M-structure,

then M is the same as the species Xn/A, as defined in below, for some A that is the

automorphism group of s.

Definition 2.3.2. Let n be a non-negative integer, A a subgroup of Sn, U and V

finite sets of cardinality n, and τ : U → V a bijection. Define a species Xn/A to be

such that

a) the set of Xn/A-structures on U is the set of (left) cosets with respect to A of

bijections [n] → U . That is,

(Xn/A)[U ] = σA | σ : [n] → U,

where σA = σ a | a ∈ A.

b) the transport

(Xn/A)[τ ] : (Xn/A)[U ] → (Xn/A)[V ]

is defined by setting

(Xn/A)[τ ](σA) = (τσ)A.

Example 2.3.3. a) There is only one element in the set En[n], so that the condition

“any two elements in En[n] are isomorphic” is automatically true. Hence the species

26


En is a molecular species, and

En =Xn

Sn.

b) We denote by Cn the species of oriented cycles of length n. We see that Cn is

a molecular species, and

Cn =Xn

Cn,

where Cn is the cyclic group of order n, i.e., Cn is the subgroup of Sn in which all

elements are n-cycles.

Remark 2.3.4. Let U be a finite set of cardinality n, and let A be a subgroup of

Sn. Then A acts naturally on the set of linear orders on U . If we call the orbit of a

linear order s on U under this A-action the A-orbit of s, then the Xn/A-structures

on U is just the set of A-orbits of the action of A on the set of linear orders on U ,

which is also, as pointed out on [2, p. 160], the quotient species of Xn by A.

The following proposition (see [15, p. 117] Example 7.4) illustrates the close con-

nection between Polya’s cycle index polynomial and the cycle index of a species.

Proposition 2.3.5. Let A be a subgroup of Sn. Then

Z(A) = ZXn/A.

Proof. Recall that the set (Xn/A)[n] is the set of cosets of A in Sn. We set

(Xn/A)[n] = g1A, g2A, . . . , gkA,

where g1, g2, . . . , gk are coset representatives. Then according to the definition of cycle

index series, we have

ZXn/A =1

n!

∑

τ∈Sn

fixXn

A[τ ] Z(τ),

27


where Z(τ) is the cycle type polynomial of τ , and the number of elements in (Xn/A)[n]

fixed by τ is

fixXn

A[τ ] = |i ∈ [k] : τgiA = giA|

=∣∣i ∈ [k] : g−1

i τgi ∈ A∣∣ .

We notice that if for some i, g−1i τgi ∈ A, then every element g in the coset represented

by giA satisfies g−1τg ∈ A. We notice also that each coset giA contains exactly |A|

elements.

Hence we get

fixXn

A[τ ] =

1

|A|

∣∣g ∈ Sn : g−1τg ∈ A∣∣ .

Therefore,

ZXn/A =1

n!

1

|A|

∑

τ∈Sn

∣∣g ∈ Sn : g−1τg ∈ A∣∣ Z(τ)

=1

n!

1

|A|

∑

τ, g∈Sng−1τg∈A

Z(τ) =1

|A|

∑

g∈Snσ∈A

1

n!Z(σ) =

1

|A|

∑

σ∈A

Z(σ) = Z(A).

Each subgroup A of Sn corresponds to a molecular species concentrated on the

cardinality n, namely, Xn/A. Proposition 2.3.5 says that Polya’s cycle index poly-

nomial of A is the same as the cycle index of the species Xn/A, hence these two

definitions are equivalent in the case of molecular species.

Example 2.3.6. There are dλ = n!/zλ permutations with cycle type λ in Sn. Thus

the cycle index polynomial of Sn, which is the same as the cycle index of the species

28


En, is

ZEn= Z(Sn) =

1

n!

∑

λ⊢n

dλpλ =∑

λ⊢n

pλ

zλ.

Example 2.3.7. Recall for any graph G, OG is the species associated to G. We

oberve that OG is indeed the molecular species Xn/ aut(G), where n is the num-

ber of vertices in G, and aut(G) is the automorphism group of G. It follows from

Proposition 2.3.5 that the cycle index of OG is the cycle index polynomial of the

automorphism group of G. That is,

ZOG= Z(aut(G)).

For example, the cycle index of the graph G in Figure 2.2 is

ZOG= Z(A),

where A is the automorphism group of G, which is the subgroup of S5 isomorphic to

S4. To be more precise,

ZOG=

1

24(p5

1 + 6p31p2 + 8p2

1p3 + 3p1p22 + 6p1p4).

We see from Definition 2.3.1 that molecular species are indecomposable under

addition. This observation leads to a molecular decomposition of any species [2,

p. 141]:

Proposition 2.3.8. Every species of structures F is the sum of its molecular

subspecies:

F =∑

M⊆FM molecular

M.

29


Example 2.3.9. The molecular decomposition of the species of graphs G is given

on [2, p. 421]:

G = 1+X +2E2 +(2E3 +2XE2)+(2E4 +2E2E2 +2XE3 +2E22 +2X2

E2 +E X2)+ · · ·

Consider the molecular species Xm/A and Xn/B. That means A is subgroup

of Sm and B is a subgroup of Sn. Yeh [31, 30] proved the following theorem for

molecular species.

Theorem 2.3.10. (Yeh) Let A be a subgroup of Sm, and B a subgroup of Sn with

n ≥ 1. We have

Xm

A·Xn

B=

Xm+n

A > B,

Xm

A

Xn

B=

Xmn

B ≀ A,

where the product group A > B acts on the set [m + n], and the wreath product group

B ≀ A, as defined by Definition 1.1.4, acts on the set [mn].

Note that Yeh’s results agree with Polya’s Theorem 1.2.6 for the cycle index

polynomials of A > B and B ≀ A.

In fact, as mentioned in Remark 2.3.4, we think of an Xm/A-structure on the set

U1 with |U1| = m as an A-orbit of linear orders on U1, and an Xn/B-structure on the

set U2 with |U2| = n as a B-orbit of linear orders on U2. Then an (Xm/A) · (Xn/B)-

structure on U = U1∨U2, where ∨ denotes the disjoint union admits an automorphism

group isomorphic to the product group A > B, as shown in Figure 2.7.

We also observe that an (Xm/A) (Xn/B)-structure is an Xm/A-structure on a

set of Xn/B-structures, which is the same as a set of B-orbits of linear orders on [n],

30


B-orbits

A-orbits

Figure 2.7. (Xm/A) · (Xn/B) = Xm+n/(A × B).

as shown in Figure 2.8, and hence admits an automorphism group isomorphic to the

wreath product B ≀ A.

B-orbits

A-orbits

B-orbits

B-orbits

Figure 2.8. (Xm/A) (Xn/B) = Xmn/(B ≀ A).

2.4. Compositional Inverse of E+

The species 1 + X is the sum of the singleton set species 1 and the species of

singletons X. A 1 + X-structure on [n] is

(1 + X)[n] =

∅, if n = 0,

1, if n = 1,

∅, otherwise.

31


The associated series of 1 + X are

(1 + X)(x) = 1 + x,

1 + X(x) = 1 + x,

Z(1+X) = 1 + p1.

A virtual species is, roughly speaking, the formal difference of species [2, p. 121].

Since the species 1 + X satisfies (1 + X)(0) = 1, Proposition 18 of [2, p. 129] gives

that there exists a unique virtual species Γ, so-called “connected (1 + X)-structures”

as defined on [2, p. 121], with

(2.4.1) 1 + X = E (Γ).

In fact, Equation (2.4.1) leads to

X = E+(Γ),

which implies that Γ is the compositional inverse of E+, written as

Γ = E〈−1〉+ (X).

The associated series of Γ are given by [2, p. 131]:

Γ(x) = log(1 + X)(x) = log(1 + x),

Γ(x) =∑

k≥1

µ(k)

klog ˜(1 + X)(xk) =

∑

k≥1

µ(k)

klog(1 + xk),

ZΓ =∑

k≥1

µ(k)

klog Z(1+X) pk =

∑

k≥1

µ(k)

klog(1 + pk),

32


where µ denotes the Mobius function, defined by

µ(k) =

0, if n has one or more repeated prime factors,

1, if n = 1,

(−1)j , if n is a product of j distinct primes.

In fact, we can compute the first several terms of the associated series of Γ:

Γ(x) =x

1!−

x2

2!+ 2

x3

3!− 6

x4

4!+ 24

x5

5!− 120

x6

6!+ 720

x7

7!− . . . ,

Γ(x) = x − x2,

(2.4.2)

ZΓ = p1 −

(1

2p2

1 +1

2p2

)+

(1

3p3

1 −1

3p3

)−

(1

4p4

1 −1

4p2

2

)+

(1

5p5

1 −1

5p5

)+ · · · .

Read [23, p. 4] derived identity (2.4.2) as the type generating series of the com-

positional inverse of the species E+.

Example 2.4.1. Let G c be the species of connected graphs, and E the species of

sets. The observation that every graph is a set of connected graphs gives rise to the

following species identity:

(2.4.3) G = E (G c),

which can be read as “a graph is a set of connected graphs”, and gives rise to the

identity

Gc = Γ(G+),

which leads to the enumeration of connected graphs:

Gc(x) = log(G (x)),

33


G c(x) =∑

k≥1

µ(k)

klog(G (xk)),

ZG c =∑

k≥1

µ(k)

klog(ZG pk − pk + 1).

Using Maple, we can compute the first several terms of the associated series of

G c:

Gc (x) =

x

1!+

x2

2!+ 4

x3

3!+ 38

x4

4!+ 728

x5

5!+ 26704

x6

6!+ 1866256

x7

7!(2.4.4)

+ 251548592x8

8!+ 66296291072

x9

9!+ . . . ,(2.4.5)

G c(x) = x + x2 + 2x3 + 6x4 + 21x5 + 112x6 + 853x7 + 11117x8 + · · · ,

ZG c = p1 +

(1

2p2

1 +1

2p2

)+

(1

3p3 +

2

3p3

1 + p1p2

)

+

(19

12p4

1 + 2p21p2 +

5

4p2

2 +2

3p1p3 +

1

2p4

)

+

(193 p3

1p2 +2

3p2p3 +

91

15p5

1 + 5p1p22 +

4

3p2

1p3 +3

5p5 + p1p4

)

+

(1669

45p6

1 +91

3p4

1p2 +38

9p3

1p3 +43

2p2

1p22 + 2p2

1p4 +8

3p1p2p3

+4

5p1p5 +

26

3p3

2 +5

2p2p4 +

25

18p2

3 +5

6p6

)+ · · · .(2.4.6)

Recall Proposition 2.3.8 that states that there is a molecular decomposition of any

species up to isomorphism. We can often identify a given species in terms of molecular

species using combinatorial operators. Figure 2.9 shows unlabeled connected graphs

on no more than 4 vertices.

34


Figure 2.9. Unlabeled connected graphs on n vertices, n ≤ 4.

In this way, the molecular decomposition of the species of connected graphs G c

takes the form

Gc = X + E2 +

(XE2 + E3

)+

(E2 X2 + XE3 + X2

E2 + E2E2 + E4 + D4

)+ · · · ,

where D4 is the molecular species corresponding to the dihedral group D4 of order 4,

that is, D4 = X4/D4, and

ZD4 = Z(D4) =1

8

(p4

1 + 2p21p2 + 3 p2

2 + 2p4

).

2.5. Multisort Species

The theory of multisort species [2, p. 100] is analogous to multivariate functions.

Definition 2.5.1. A k-set U is a k-tuple of sets

U = (U1, U2, . . . , Uk).

A multifunction f from (U1, U2, . . . , Uk) to (V1, V2, . . . , Vk), denoted

f : (U1, U2, . . . , Uk) → (V1, V2, . . . , Vk),

35


is a k-tuple of functions f = (f1, f2, . . . , fk) such that fi : Ui → Vi for i = 1, . . . , k.

Such f is called bijective if each fi is a bijection.

Let Bk be the category of k-sets with bijective multifunctions.

Definition 2.5.2. A species of k sorts, where k ≥ 1, is a functor from the category

of finite k-sets Bk to the category of finite sets B, i.e.,

F : Bk → B.

A k-sort species F gives rise for each k-set

U = (U1, U2, . . . , Uk)

a finite set F [U1, U2, . . . , Uk], and for each bijective multifunction

σ = (σ1, σ2, . . . , σk) : (U1, U2, . . . , Uk) → (V1, V2, . . . , Vk)

a bijection

F [σ] = F [σ1, σ2, . . . , σk] : F [U1, U2, . . . , Uk] → F [V1, V2, . . . , Vk]

that is called the transport of F -structures along σ.

We denote by F (X, Y ) a 2-sort species F . We use the notation [m, n], where m

and n are positive integers, to denote the 2-set (1, 2, 3, . . . , m, 1, 2, 3, . . . , n).

The exponential generating series of F (X, Y ) is

F (x, y) =∑

m,n≥0

∣∣F [m, n]∣∣ xm

m!

yn

n!,

where F [m, n] = F [1, 2, . . . , m, 1, 2, . . . , n] is the number of labeled F -structures

on the 2-set [m, n].

36


F

Figure 2.10. Multisort species.

The type generating series of F (X, Y ) is

F (x, y) =∑

m,n≥0

fm,n xmyn,

where fm,n is the number of unlabeled F -structures in which the cardinality of the

first sort set is m and the cardinality of the second sort set is n.

The cycle index of F (X, Y ) is

ZF (X,Y ) =∑

m,n≥0

( ∑

λ⊢m, µ⊢n

fix F [λ, µ]pλ[x]

zλ

pµ[y]

zµ

),

where pλ[x] and pλ[y] are power sum symmetric functions in variable sets x1, x2, . . .

and y1, y2, . . . , respectively, fix F [λ, µ] denotes the number of F -structures on [m, n]

fixed by F [σ, π], where σ is a permutation of [m] with cycle type λ and π is a permu-

tation of [n] with cycle type µ.

Example 2.5.3. Let A (X, Y ) be the 2-sort species of rooted trees whose internal

vertices are of sort X and whose leaves are of sort Y . Figure 2.11 shows the transport

37


of structures of two A (X, Y )-structures under a bijective multifunction

σ =

(2

j

3

a

4

g

5

m

7

f

8

e

9

d

12

n

∣∣∣∣1

l

6

k

10

h

11

b

13

i

14

c

):

A [σ]mf

e

ch

g

n

d

k bi

j

a

l

4

12

9

6 1113

2

57

8

1410 1

3

Figure 2.11. The transport of structures A [σ] sends an element ofA [U1, U2] to an element of A [V1, V2], where

U1 = 2, 3, 4, 5, 7, 8, 9, 12, U2 = 1, 6, 10, 11, 13, 14,V1 = j, a, g, m, f, e, d, n, V2 = l, k, h, b, i, c.

2.6. Arithmetic Product of Species

The arithmetic product was studied by Manuel Maia and Miguel Mendez [18].

They developed a decomposition of a set, called a rectangle, which, as we shall see, is

essentially an equivalence class of high dimensional linear orders.

Definition 2.6.1. Let U be a finite set. Suppose |U | = ab. A rectangle on U of

height a is a pair (π1, π2) such that π1 is a partition of U with a blocks, each of size

b, and π2 is a partition of U with b blocks, each of size a, and if B is a block of π1

and B′ is a block of π2 then |B ∩ B′| = 1.

38


Figure 2.12 shows two representations of a rectangle on [12] of height 3. They

both represent the rectangle (π1, π2).

π1 = 1, 3, 4, 9, 2, 5, 8, 10, 6, 7, 11, 12,

π2 = 1, 8, 12, 3, 5, 6, 2, 4, 11, 7, 9, 10.

4

4

611712

1 9 3

52108

1

28

12 11

3

5

6

9

10

7

∼

Figure 2.12. A rectangle on [12] of height 3.

Equivalently we can represent a rectangle as a bipartite graph. A rectangle which

is the same as in Figure 2.12 is shown by Figure 2.13, where the labels are on the

10

9 8 12

4

3

1

2

5

711

6

π1 = A1, A2, A3

B1

B2

B3

B4

A1

A2

A3 π2 = B1, B2, B3, B4

Figure 2.13. A rectangle on [12] of height 3.

edges, and the vertex sets on the left and on the right represent partitions π1 and π2.

More precisely, each edge x in the bipartite graph corresponds to exactly one pair of

vertices (Ai, Bj) with Ai ∈ π1 and Bj ∈ π2, and this can be interpreted as x being

the only element in the intersection of Ai and Bj .

39


Definition 2.6.2. Let U be a finite set. A k-rectangle on U is a k-tuple of partitions

(π1, π2, . . . , πk) such that

a) for each i ∈ [k], πi has ai blocks, each of size |U |/ai, where |U | =k∏

i=1

ai.

b) for any k-tuple (B1, B2, . . . , Bk), where Bi is a block of πi for each i ∈ [k],

we have |B1 ∩ B2 ∩ · · · ∩ Bk| = 1.

Figure 2.14 shows a 3-rectangle (π1, π2, π3), represented by a 3-partite graph, and

labeled are on the triangles.

C1

π3 = C1, C2

π1 = A1, A2, A3, A4

A4A3A2A1

C2

B3

B2

B1π2 = B1, B2, B3

Figure 2.14. A 3-rectangle labeled on the triangles.

We denote by N the species of rectangles, and by N (k) the species of k-rectangles.

Let n =∏k

i=1 ai, and let ∆ be the set of bijections of the form

δ : [a1] × [a2] × · · · × [ak] → [n].

Note that the cardinality of the set ∆ is n!. The group

k∏

i=1

Sai= σ = (σ1, σ2, . . . , σk) : σi ∈ Sai

acts on the set ∆ by setting

(σ · δ)(i1, i2, . . . , ik) = δ(σ1(i1), σ2(i2), . . . , σk(ik)),

40


for each (i1, i2, . . . , ik) ∈ [a1] × [a2] × · · · × [ak]. We observe that this group action

result in a set of∏k

i=1 Sai-orbits, and that each orbit consists of exactly a1!a2! · · ·ak!

elements of ∆. Observe further that there is a one-to-one correspondence between the

set of∏k

i=1 Sai-orbits on the set ∆ and the set of k-rectangles of the form (π1, . . . , πk),

where each πi has ai blocks. Therefore, the number of such k-rectangles is

n

a1, a2, . . . , ak

:=

n!

a1!a2! · · ·ak!.

Definition 2.6.3. Let F1 and F2 be species of structures with F1[∅] = F2[∅] = ∅.

The arithmetic product of F1 and F2, denoted F1 ⊡ F2, is defined by setting for each

finite set U ,

(F1 ⊡ F2)[U ] =∑

(π1 π2)∈N [U ]

F1[π1] × F2[π2],

where the sum represents the disjoint union. In other words, an F1 ⊡ F2-structure on

a finite set U is a tuple of the form ((π1, f1), (π2, f2)), where fi is an Fi-structure on

the blocks of πi for each i.

A bijection σ : U → V sends a partition π of U to a partition π′ of V , namely,

σ(π) = π′ = σ(B) : B is a block of π. Thus σ induces a bijection σπ : π → π′,

sending each block of π to a block of π′.

The transport of structures for any bijection σ : U → V is defined by

(F1 ⊡ F2)[σ]((π1, f1), (π2, f2)) = ((π′1, F1[σπ1 ](f1)), (π′

2, F2[σπ2 ](f2))).

The arithmetic product on species satisfies the following properties. They are

straightforward to check using the definition.

41


F2F1 F2 F1

X

X

X

X

X

X

X

X

X

XX

X

XX

X

Figure 2.15. Arithmetic product F1 ⊡ F2.

Proposition 2.6.4. (Maia and Mendez) Let F1, F2 and F3 be species of structures

with Fi[∅] = ∅ for i = 1, 2, 3, and let X be the species of singleton sets. Then we have

(commutativity) F1 ⊡ F2 = F2 ⊡ F1,

(associativity) F1 ⊡ (F2 ⊡ F3) = (F1 ⊡ F2) ⊡ F3,

(distributivity) F1 ⊡ (F2 + F3) = F1 ⊡ F2 + F1 ⊡ F3,

(unit) F1 ⊡ X = X ⊡ F1 = F1.

FF

X

X

X

X

X

X X

Figure 2.16. X is the unit of the arithmetic product: F ⊡ X = F .

The following proposition about the arithmetic product of two molecular species

could be taken as an alternative definition of the arithmetic product.

Proposition 2.6.5. Let Xm/A and Xn/B be two molecular species. That is, A

is a subgroup of Sm, and B is a subgroup of Sn. Then

Xm

A⊡

Xn

B=

Xmn

A × B,

42


where A×B is the product group acting on the set [mn] as defined by Definition 1.1.3.

Example 2.6.6. Recall that the molecular species E2 is the same as the species

X2/S2. The arithmetic product of the species E2 with itself is a species concentrated

on the cardinality 4. To be more precise, the set of E2 ⊡ E2-structures on a set U

with |U | = 4 are obtained by taking the S2 × S2-orbits of 2 by 2 squares filled with

elements of U , where the group S2×S2 acts by switching the rows and the columns.

In other words, E2 ⊡ E2 = X4/(S2 × S2).

E2

a b

dc

E2

Figure 2.17. The species E2 ⊡ E2 is isomorphic to the speciesX4/(S2 × S2).

We can quickly verify Proposition 2.6.5 by observing that the Xm/A-structures

are A-orbits of linear orders on [m], and the Xn/B-structures are B-orbits of linear

orders on [n], and hence an (Xm/A)⊡(Xn/B)-structure on [mn] is just a matrix whose

rows and columns are A-orbits of linear orders on [m] and B-orbits of linear orders on

[n], respectively. The automorphism group of such an (Xm/A) ⊡ (Xn/B)-structure

is therefore isomorphic to the product group A × B.

The associativity of the arithmetic product allows us to talk about the arithmetic

product of a set of species.

43


B-orbits

A-orbits

Figure 2.18. (Xm/A) ⊡ (Xn/B) = Xmn/(A × B).

Definition 2.6.7. The arithmetic product of species F1, F2, . . . , Fk with Fi(∅) = ∅

for all i is defined by setting

k

⊡i=1

Fi = F1 ⊡ F2 ⊡ · · ·⊡ Fk,

which sends each finite set U to the set

k

⊡i=1

Fi[U ] =∑

F1[π1] × F2[π2] × · · · × Fk[πk],

where the sum is taken over all k-rectangles (π1, π2, . . . , πk) of U , and represents the

disjoint union. We denote by F ⊡k the arithmetic product of k copies of F .

For each bijection σ : U → V , the transport of structures ofk

⊡i=1

Fi along σ sends

ank

⊡i=1

Fi-structure on U of the form

((π1, f1), (π2, f2), . . . , (πk, fk))

to andk

⊡i=1

Fi-structure on V of the form

((π′1, F1[σπ1 ]f1), (π′

2, F2[σπ2 ]f2), . . . , (π′k, Fk[σπk

]fk)),

44


where σπiis the bijection induced by σ sending blocks of πi to blocks of π′

i.

Proposition 2.6.8. (Maia and Mendez) We have the following identities for the

species of k-rectangles.

N = E+ ⊡ E+,

N(k) = E

⊡k+ ,

N(i+j) = N

(i)⊡ N

(j).

Proof. These identities follow straightforwardly from the definitions of rectangle

and arithmetic product of species.

Figures 2.17 and 2.19 show that the set E2⊡E2[4] is the same as the set of rectangles

on [4] of height 2.

E2

1 2

43

E2 E2E2

4

1 3

2

21

4 3E2E2E2

1 3

2 4

E2 E2E2

23

1 4 1 4

32E2E2

Figure 2.19. The E2 ⊡ E2-structures on the set [4].

Maria and Mendez [18] gave the formula for the cycle index of the arithmetic

product.

45


Theorem 2.6.9. (Maria and Mendez) Let species F1 and F2 satisfy

F1[∅] = F2[∅] = ∅.

Then we have

(2.6.1) ZF1⊡F2 = ZF1 ⊠ ZF2,

where the operation ⊠ on the power sum symmetric functions on the right-hand side

of the equation is given by Definition 1.2.3.

We notice that due to the linearities of both the arithmetic product of species and

the operation ⊠ on power sum symmetric functions, identity (2.6.1) reduces to the

case when F1 and F2 are both molecular species.

Suppose, say, F1 = Xm/A and F2 = Xn/B for some integers m, n, and some

A and B subgroups of Sm and Sn, respectively. We apply Proposition 2.6.5 and

Proposition 2.3.5 to translate the formula

Z(Xm/A)⊡(Xn/B) = ZXm/A ⊠ ZXn/B

into

(2.6.2) Z(A × B) = Z(A) ⊠ Z(B).

Identity (2.6.2) can be roughly verified by observing that if a is an element of A

and b is an element of B, then the cycle type of (a, b) in the group A × B satisfies

pc.t.((a,b)) = pc.t.(a) ⊠ pc.t.(b).

46


This is because each pair of a k-cycle in a and an l-cycle in b generates cycles of

length lcm(k, l) in the group A × B with multiplicity gcd(k, l), and that the formula

for such pairs of k-cycles in a and l-cycles in b is

pikk ⊠ pjl

l = pgcd(k,l) ikjl

lcm(k,l) .

Figure 2.20 gives an illustration of p3 ⊠ p4 = p12.

Figure 2.20. A 3-cycle in the group A and a 4-cycle in the group Bgenerates a 12-cycle in the group A × B.

Summing up on all different cycle lengths of a and b, we get the desired result.

47

CHAPTER 3

Cartesian Product of Graphs and Prime Graphs

3.0. Introduction

In this chapter, we look at the well-known operation of Cartesian product (Defini-

tion 3.1.1) on graphs, and the notion of prime graphs (Definition 3.1.3) with respect

to Cartesian multiplication. Our goal is to find the cycle index of the species of prime

graphs.

First, we get a relation (Proposition 3.1.8) between the Cartesian product of

graphs and the arithmetic product of species (Definition 2.6.3). We then use the

tool of Dirichlet exponential generating series (Definition 3.2.3) to get a formula

(Theorem 3.2.9) expressing the number of labeled prime graphs in terms of the number

of labeled connected graphs. Furthermore, we see that the unique factorization of a

nontrivial connected graph into the product of powers of prime graphs (Sabidussi [26])

gives a free commutative monoid structure on the set of unlabeled connected graphs,

and leads to a formula (Theorem 3.3.10) that counts unlabeled prime graphs in terms

of unlabeled connected graphs.

In terms of species, the arithmetic product of species studied by Maia and Mendez [18]

comes back into play. As it turns out, we need a stronger notion than arithmetic

product in order to express the species of connected graphs in terms of the species

of prime graphs. We therefore define a new operation, the exponential composition

of species (Definition 3.5.1), which is related to the arithmetic product of species as

48

CHAPTER 3. PRIME GRAPHS

the composition of species is related to the multiplication of species, and get a for-

mula (Theorem 3.6.2) expressing the species of connected graphs as the exponential

composition of the species of prime graphs. An explicit formula for the inverse of

exponential composition would be nice to find, but that problem remains open.

The enumeration of the species of prime graphs is therefore complete by applying

the enumeration theorem (Theorem 3.4.4) for the exponential composition of species,

which is a generalization of the enumeration theorem by Palmer and Robinson about

the cycle index polynomial of the exponentiation group (Theorem 1.3.6).

3.1. Cartesian Product of Graphs

For any graph G, we let V (G) be the vertex set of G, E(G) the edge set of G,

and l(G) = |V (G)| the number of vertices in G. Two graphs G and H with the

same number of vertices are said to be isomorphic, denoted G ∼= H , if there exists a

bijection from V (G) to V (H) that preserves adjacency. Such a bijection is called an

isomorphism from G to H . In the case when G and H are identical, this bijection

is called an automorphism of G. The collection of all automorphisms of G, denoted

aut(G), constitutes a group called the automorphism group of G.

We call the isomorphism classes of graphs unlabeled graphs. If G is a graph with

n vertices, L(G) is the number of graphs isomorphic to G with vertex set [n]. It is

well-known that

L(G) =l(G)!

|aut(G)|.

We use the notation∑n

i=1 Gi = G1 + G2 + · · ·+ Gn to mean the disjoint union of

a set of graphs Gii=1,...,n.

Definition 3.1.1. The Cartesian product of graphs G1 and G2, denoted G1 ⊙ G2,

as defined by Sabidussi [26] under the name the weak Cartesian product, is the graph

49


whose vertex set is

V (G1 ⊙ G2) = V (G1) × V (G2) = (u, v) : u ∈ V (G1), v ∈ V (G2),

in which (u, v) is adjacent to (w, z) if

either u = w and v, z ∈ E(G2) or v = z and u, w ∈ E(G1).

For simplicity and without ambiguity, we call G1 ⊙G2 the product of G1 and G2.

2

3,3’

4,3’3,2’

2,2’

2,3’

1,3’

1,2’3,1’

4,1’2,1’

1,1’

1’

2’3’

1

3

4

4,2’

Figure 3.1. The Cartesian product of a graph with vertex set1, 2, 3, 4 and a graph with vertex set 1′, 2′, 3′ is a graph with vertexset (i, j), where i ∈ [4] and j ∈ [3]′.

The following properties of the Cartesian multiplication can be verified straight-

forwardly.

Proposition 3.1.2. Let G1, G2 and G3 be graphs. Then

(commutativity) G1 ⊙ G2∼= G2 ⊙ G1,

(associativity) (G1 ⊙ G2) ⊙ G3∼= G1 ⊙ (G2 ⊙ G3).

50


These properties allow us to talk about the Cartesian product of a set of graphs

Gii∈I , denoted ⊙i∈I

Gi. We denote by Gn the Cartesian product of n copies of G.

Definition 3.1.3. A graph G is prime with respect to Cartesian multiplication if G

is a connected graph with more than one vertex such that G ∼= H1 ⊙H2 implies that

either H1 or H2 is a singleton vertex.

Two graphs G and H are called relatively prime with respect to Cartesian mul-

tiplication, if and only if G ∼= G1 ⊙ J and H ∼= H1 ⊙ J imply that J is a singleton

vertex.

We denote by P the species of prime graphs. We also denote by C the set of

unlabeled connected graphs, and by P the set of unlabeled prime graphs.

If G is a connected graph, then G can be decomposed into prime factors, that is,

there is a set Pii∈I of prime graphs such that G ∼= ⊙i∈I

Pi. In Figure 3.2, a connected

graph G with 24 vertices is decomposed into the product of prime graphs P1, P2, P3

with 3, 2 and 4 vertices, respectively.

=

Figure 3.2. The decomposition of a connected graph into prime graphs.

Furthermore, we have the following theorem given by Sabidussi [26].

Theorem 3.1.4. (Sabidussi) For any non-trivial connected graph G, the factor-

ization of G into the Cartesian product of prime powers is unique up to isomorphism.

51


The automorphism groups of the Cartesian product of a set of graphs was studied

by Sabidussi [26] and Palmer [19].

Theorem 3.1.5. (Sabidussi) Let Gii=1,...,n be a set of graphs. Then the auto-

morphism group of the Cartesian product of Gii=1,...,n is isomorphic to the auto-

morphism group of the sum of Gii=1,...,n. That is,

aut

(n⊙i=1

Gi

)∼= aut

( n∑

i=1

Gi

).

Theorem 3.1.6. (Sabidussi) Let G1, G2, . . . , Gn be connected graphs which are

pairwise relatively prime with respect to Cartesian multiplication. Then the automor-

phism group of the Cartesian product of Gii=1,...,n is isomorphic to the product of

each of the automorphism groups of Gii=1,...,n. That is,

(3.1.1) aut

(n⊙i=1

Gi

)∼=

n∏

i=1

aut(Gi).

Example 3.1.7. The prime graphs P1, P2 and P3 in Figure 3.2 are pairwise relatively

prime, so the automorphism group of G is

aut(G) = aut(P1) × aut(P2) × aut(P3)

= S2 × S2 × V4,

where V4∼= S2 × S2.

Recall the arithmetic product of species defined by Definition 2.6.3 and the species

associated to a graph defined in Example 2.1.4. We present in the following the close

connection between the arithmetic product and the Cartesian product.

52


Proposition 3.1.8. Let G1 and G2 be two graphs that are relatively prime to each

other. Then the species associated to the Cartesian product of G1 and G2 is equivalent

to the arithmetic product of the species associated to G1 and the species associated to

G2. That is,

OG1⊙G2 = OG1 ⊡ OG2

Proof. Let l(G1) = m and l(G2) = n. Then l(G1 ⊙ G2) = mn.

Since G1 and G2 are relatively prime, applying Theorem 3.1.6 we get

aut(G1 ⊙ G2) = aut(G1) × aut(G2).

Therefore,

OG1⊙G2 =X l(G1⊙G2)

aut(G1 ⊙ G2)=

Xmn

aut(G1) × aut(G2)

=Xm

aut(G1)⊡

Xn

aut(G2)= OG1 ⊡ OG2 .

Note that if G1 and G2 are not relatively prime to each other, then the species

associated to the Cartesian product of G1 and G2 is generally different from the

arithmetic product of OG1 and OG2 . This is because that the automorphism group

of the product of the graphs is no longer the product of the automorphism groups of

the graphs. For example, we will see in Remark 3.2.2 that for any prime graph P ,

aut(P ⊙ P ) = aut(P )S2,

and aut(P )S2 is generally much larger than aut(P )2.

53


3.2. Labeled Prime Graphs

We introduce in the following an important theorem by Sabidussi about the au-

tomorphism group of a connected graph using its prime factorization.

Theorem 3.2.1. (Sabidussi) Let G be a connected graph with prime factorization

G ∼= P s11 ⊙ P s2

2 ⊙ · · · ⊙ P sk

k ,

where for r = 1, 2, . . . , k, all Pr are distinct prime graphs, and all sr are positive

integers. Then we have

aut(G) ∼=

k∏

r=1

aut(P sr

r ) ∼=

k∏

r=1

aut(Pr)Ssr .

In other words, the automorphism group of G is generated by the automorphism groups

of the factors and the transpositions of isomorphic factors.

Remark 3.2.2. (A quick verification of Theorem 3.2.1, and more.) Note that the

P srr , for r = 1, 2, . . . , k, are pairwise relatively prime. With Theorem 3.1.6 handy, we

see that Theorem 3.2.1 is equivalent to the following special case:

If P is any prime graph and k is a nonnegative integer, then the auto-

morphism group of P k is the exponentiation group aut(P )Sk , i.e.,

aut(P k) = aut(P )Sk.

In fact, we quickly observe that aut(P )Sk is a subgroup of aut(P k), since every

element

(α, τ) = (α; τ(1), τ(2), . . . , τ(k)) ∈ aut(P )Sk

54


with α ∈ Sk and τ(i) ∈ aut(P ), for all i, gives rise to an automorphism of the product

graph P k by letting each τ(i) act on a copy of P in P k and letting α act on the k

copies of P in P k. More precisely, since each of the vertices of P k is a k-vector where

the i-th coordinate is contributed by a vertex in the i-th copy of P , the permutation

α acts on P k by permuting the coordinates of the vertices of P k. Therefore, the

exponentiation group aut(P )Sk is a subgroup of the automorphism group of P k.

Hence what Sabidussi’s theorem really says is that the automorphism group of P k

is no bigger than the exponentiation group aut(P )Sk . Therefore, Theorem 3.2.1 can

be verified by showing that these two groups contain the same number of elements.

That is, it remains to show that

aut(P k) = | aut(P )Sk |.

Recall that for any graph G, kG is the sum of k copies of G. By Theorem 3.1.5,

we have

| aut(P k)| = | aut(kP )|.

Therefore, it suffices for us to show that

| aut(kP )| = k! | aut(P )|k,

where the right-hand side is the same as the order of the exponentiation group

aut(P )Sk .

Note that kP has k connected components, each of which is a copy of P . Let

α be an automorphism of kP , and let v1, v2 be vertices of kP that are in the same

connected component. Since automorphisms preserve adjacency, α(v1) and α(v2) are

in the same connected component of α(kP ) = kP . Therefore, α sends one connected

component of kP to another connected component of kP , possibly the same one.

55


S3

Figure 3.3. The automorphisms of kP are generated by elements inaut(P ) and Sk.

Let us assume, say, that α(P1) = P2, where P1 and P2 are two arbitrary connected

components of kP , possibly the same one. This means that P1 and P2 are both copies

of P . Let α1 be the action of α restricted to P1. Then there is some σ : P2 → P1

such that σα1 is an automorphism of P . Therefore, such an α can be obtained by

taking an automorphism of each connected component of kP , separately, followed by

putting a permutation on [k] over the k copies of P . In the meanwhile, the above

argument also implies that this is the only way to construct an automorphism of kP .

Thus we see from Definition 1.1.4 that the automorphism group of kP is the wreath

product group aut(P ) ≀ Sk, i.e.,

aut(kP ) = aut(P ) ≀ Sk,

which gives that

| aut(kP )| = | aut(P ) ≀ Sk| = k! | aut(P )|k.

In what follows in this section all graphs considered are connected.

Definition 3.2.3. The Dirichlet exponential generating series for a sequence of num-

bers ann∈N is defined by∑n≥1

an/n! ns.

56


Multiplication of Dirichlet exponential generating series is given by

(3.2.1)

(∑

n≥1

an

n! ns

)(∑

n≥1

bn

n! ns

)=

∑

n≥1

cn

n! ns,

where

cn =∑

k|n

n

k

akbn/k =

∑

k|n

n!

k! (n/k)!akbn/k.

The Dirichlet exponential generating function for a species F with the restriction

F [∅] = ∅ is defined by

D(F ) =∑

n≥1

|F [n]|

n! ns.

Example 3.2.4. Recall that for any graph G, OG is the species associated to G.

We denote by D(G) the Dirichlet exponential generating function for the species OG.

More explicitly,

D(G) = D(OG) =L(G)

l(G)! l(G)s.

It is illustrated by the following proposition that the Dirichlet exponential gener-

ating functions are useful for enumeration involving the arithmetic product of species.

Proposition 3.2.5. (Maia and Mendez) Let F1 and F2 be species with Fi[∅] = ∅

for i = 1, 2. Then

(3.2.2) D(F1 ⊡ F2) = D(F1) D(F2).

Proof. This proof was given by Maia and Mendez [18, p. 6]. We calculate the

number of F1 ⊡ F2-structures on the set [n]:

|F1 ⊡ F2[n]| =∑

(π1,π2)∈N [n]

|F1[π1]| |F2[π2]|

57


=∑

a|n

∑

(π1,π2)∈N [n]height(π1,π2)=a

|F1[a]| |F2[n/a]|

=∑

a|n

n

a

|F1[a]| |F2[n/a]|.

Equation (3.2.2) follows from the multiplication rule of Dirichlet exponential gen-

erating functions given by Equation (3.2.1).

Lemma 3.2.6. Let G1 and G2 be relatively prime graphs. Then

D(G1 ⊙ G2) = D(G1)D(G2)

Proof. Applying Proposition 3.1.8 and Proposition 3.2.5, we get

D(G1 ⊙ G2) = D(OG1⊙G2) = D(OG1 ⊡ OG2) = D(OG1)D(OG2) = D(G1)D(G2).

Lemma 3.2.7. Let P be any prime graph. Let T be the sum of all nonnegative

integer powers of P , i.e., T =∑

k≥0 P k. Then the Dirichlet exponential generating

functions for T and P are related by

D(T ) = exp(D(P )).

Proof. For any graph G, we have

L(G) =l(G)!

|aut(G)|,

and

D(G) =L(G)

l(G)! · l(G)s=

1

|aut(G)| · l(G)s.

58


Now it follows from Remark 3.2.2 that

∣∣aut(P k)∣∣ =

∣∣aut(P )Sk∣∣ = k! · |aut(P )|k ,

which gives that

L(P k) =l(P k)!

| aut(P k)|=

l(P k)!

k! · |aut(P )|k.

Hence we get

D(P k) =L(P k)

l(P k)! · l(P k)s=

1

k! · |aut(P )|k · l(P )ks=

D(P )k

k!.

Summing up on k, we get

D(T ) =∑

k≥0

D(P )k

k!= exp(D(P )).

Example 3.2.8. Let P be the species of prime graphs, and G c the species of con-

nected graphs. Then D(G c) and D(P) are the Dirichlet exponential generating

functions for these two species, respectively:

D(G c) =∑

n≥1

|G c[n]|

n! ns=

∑

G∈C

D(G), D(P) =∑

n≥1

|P[n]|

n! ns=

∑

P∈P

D(P ),

where C is the set of unlabeled connected graphs and P is the set of unlabeled prime

graphs.

Theorem 3.2.9. Let G c be the species of connected graphs, and let P be the

species of prime graphs. We have

D(G c) = exp (D(P)).

59


Proof. Lemma 3.2.6 gives that the Dirichlet exponential generating function of

a product of two relatively prime graphs is the product of the Dirichlet exponential

generating functions of the two graphs. Note that according to Proposition 3.1.2, the

operation of Cartesian product on graphs is associative up to isomorphism. Then it

follows that if we have a set of pairwise relatively prime graphs Gii=1,2,...,r, and let

G =r⊙i=1

Gi, then

D(G) =

r∏

i=1

D(Gi).

Now according to the definition of the Dirichlet exponential generating function

for graphs, we get

D(G c) =∑

G∈C

D(G) =∏

P∈P

D

(∑

k≥0

P k

)=

∏

P∈P

exp(D(P ))

= exp(∑

P∈P

D(P ))

= exp(D(P)).

Recall the exponential generating series of G c given by (2.4.5):

Gc (x) =

x

1!+

x2

2!+ 4

x3

3!+ 38

x4

4!+ 728

x5

5!+ 26704

x6

6!+ 1866256

x7

7!

+ 251548592x8

8!+ 66296291072

x9

9!+ . . . ,

We obtain D(G c) by replacing xn with n−s for each n in the above expression:

D(G c) =∑

n≥1

|G c[n]|1

n! ns

=1

1! 1s+

1

2! 2s+ 4

1

3! 3s+ 38

1

4! 4s+ 728

1

5! 5s+ 26704

1

6! 6s+ 1866256

1

7! 7s

+ 2515485921

8! 8s+ 66296291072

1

9! 9s+ . . . .

60


Theorem 3.2.9 gives a way of counting labeled prime graphs by writing

D(P) = log D(G c).

For example, we write down the first terms of D(P) as follows:

D(P) =1

1! 1s+

1

2! 2s+ 4

1

3! 3s+ 35

1

4! 4s+ 728

1

5! 5s+ 26464

1

6! 6s+ 1866256

1

7! 7s

+ 2515183521

8! 8s+ 66296210432

1

9! 9s+ . . . .

Let prn and cn be the number of labeled prime graphs and the number of labeled

connected graphs, respectively. We obtain the following table.

Table 1. Values of prn and cn, for n ≤ 12.

n prn cn

1 1 12 1 13 4 44 35 385 728 7286 26464 267047 1866256 18662568 251518352 2515485929 66296210432 66296291072

10 34496477587456 3449648859481611 35641657548953344 3564165754895334412 73354596197458024448 73354596206766622208

3.3. Unlabeled Prime Graphs

In this section all graphs considered are unlabeled and connected.

Definition 3.3.1. The (formal) Dirichlet series of a sequence ann=1,2,...,∞ is de-

fined to be∑∞

n=1 an/ns.

61


The multiplication of Dirichlet series is given by

∑

n≥1

an

ns·∑

m≥1

bn

ns=

∑

n≥1

(∑

k|n

akbn/k

)1

ns.

Definition 3.3.2. A monoid is a semigroup with a unit. A free commutative monoid

is a commutative monoid M with a set of primes P ⊆ M such that each element m ∈

M can be uniquely decomposed into a product of elements in P up to rearrangement.

Let M be a free commutative monoid. We get a monoid algebra CM , in which

the elements are all formal sums∑

m∈M cmm, where cm ∈ C, with addition and

multiplication defined naturally.

For each m ∈ M , we associate a length l(m) that is compatible with the multipli-

cation in M . That is, for any m1, m2 ∈ M , we have l(m1)l(m2) = l(m1m2).

It is well-known that the monoid algebra yields the following identity:

Proposition 3.3.3. Let M be a free commutative monoid with prime set P . The

following identity holds in the monoid algebra CM :

∑

m∈M

m =∏

p∈P

1

1 − p.

Furthermore, we can define a homomorphism from M to the ring of Dirichlet series

under which each m ∈ M is sent to 1/l(m)s, where l is a length function of M .

Therefore,∑

m∈M

1

l(m)s=

∏

p∈P

1

1 − l(p)−s.

Example 3.3.4. Let N denote the set of all natural numbers, and let P denote the

set of all prime numbers. Then N is a free commutative monoid with prime set P,

and the length function is given by l(n) = n for all n ∈ N. As an application of

Proposition 3.3.3, we have the following well-known identity for expressing the zeta

62


function:

ζ(s) =∑

n∈N

1

ns=

∏

p∈P

1

1 − p−s.

Recall that C is the set of unlabeled connected graphs under the operation of

Cartesian product. The unique factorization theorem of Sabidussi gives C the struc-

ture of a commutative free monoid with a set of primes P, where P is the set of

unlabeled prime graphs. This is saying that every element of C has a unique fac-

torization of the form be11 be2

2 · · · bek

k , where the bi are distinct primes in P. Let l(G),

the number of vertices in G, be a length function for C. We have the following

proposition.

Lemma 3.3.5. Let C and P be the set of unlabeled connected graphs and the set of

unlabeled prime graphs, respectively. We have

∑

G∈C

1

l(G)s=

∏

P∈P

1

1 − l(P )−s.

The enumeration of prime graphs was studied by Raphael Bellec [1]. We use

Dirichlet series to count unlabeled connected prime graphs.

Theorem 3.3.6. Let cn be the number of unlabeled connected graphs on n vertices,

and let bm be the number of unlabeled prime graphs on m vertices. Then we have

(3.3.1)∑

n≥1

cn

ns=

∏

m≥2

1

(1 − m−s)bm.

Furthermore, if we define numbers dn for positive integers n by

(3.3.2)∑

n≥1

dn

ns= log

∑

n≥1

cn

ns,

63


then

(3.3.3) dn =∑

ml=n

bm

l,

where the sum is over all pairs (m, l) of positive integers with ml = n.

Remark 3.3.7. A quick observation from Equation (3.3.3) is that bn = dn whenever

n is not of the form rk for some k > 1.

In what follows, we introduce an interesting recursive formula for computing dn.

To start with, we differentiate both sides of Equation (3.3.2) with respect to s and

simplify. We get that

∑

n≥2

log ncn

ns=

(∑

n≥1

cn

ns

)(∑

n≥2

log ndn

ns

),

which gives

(3.3.4) cn log n =∑

ml=n

cmdl log l.

Since c1 is the number of connected graphs on 1 vertex, c1 = 1. It follows from

Equation (3.3.4) easily that dp = cp when p is a prime number. Therefore, if p is a

prime number, bp = dp = cp. This fact can be seen directly, since a connected graph

with a prime number of vertices is a prime graph.

Raphael Bellec used Equation (3.3.4) to find formulae for dn where n is a product

of two different primes or a product of three different primes:

If n = pq where p 6= q,

(3.3.5) dn = cn − cpcq;

64


If n = pqr where p, q and r are distinct primes,

(3.3.6) dn = cn + 2cpcqcr − cpcqr − cqcpr − crcpq.

In fact, Equations (3.3.5) and (3.3.6) are special cases of the following proposition.

Proposition 3.3.8. Let dn, cn be defined as above. Then we have

dn = cn −1

2

∑

n1n2=n

cn1cn2 +1

3

∑

n1n2n3=n

cn1cn2cn3 − . . . .

Proof. We can use the identity

log(1 + x) = x −1

2x2 +

1

3x3 −

1

4x4 + . . .

to compute from Equation (3.3.2) that

∑

n≥1

dn

ns= log

(1 +

∑

n≥2

cn

ns

)

=∑

n≥2

cn

ns−

1

2

(∑

n≥2

cn

ns

)2

+1

3

(∑

n≥2

cn

ns

)3

− . . . .

Equating coefficients of n−s on both sides, we get the desired result.

Proof of Theorem 3.3.6. We start with

∑

m

1

l(m)s=

∏

p

1

1 − l(p)−s,

where the left-hand side is summed over all connected graphs, and the right-hand side

is summed over all prime graphs. Regrouping the summands on the left-hand side with

respect to the number of vertices in m, we get the left-hand side of Equation (3.3.1).

Regrouping the factors on the right-hand side with respect to the number of vertices

in p, we get the right-hand side of Equation (3.3.1).

65


Taking the logarithm of both sides of Equation (3.3.1), we get

log∑

n≥1

cn

ns= log

∏

m≥2

1

(1 − m−s)bm=

∑

m≥2

bm log1

1 − m−s

=∑

m≥2

(bm

∑

l≥1

m−sl

l

)=

∑

m≥2, l≥1

bm

l msl,

and Equation (3.3.3) follows immediately.

Next, we will compute the numbers bn in terms of the numbers dn using the

following lemma.

Lemma 3.3.9. Let Dii=1,... and Jii=1,... be sequences of numbers satisfying

(3.3.7) Dk =∑

l|k

Jk/l

l,

and let µ be the Mobius function. Then we have

Jk =1

k

∑

l|k

µ

(k

l

)lDl.

Proof. Multiplying by k on both sides of Equation (3.3.7) , we get

kDk =∑

l|k

k

lJk/l =

∑

l|k

lJl.

Applying the Mobius inversion formula, we get

kJk =∑

l|k

µ

(k

l

)lDl.

Therefore,

Jk =1

k

∑

l|k

µ

(k

l

)lDl.

66


Given any natural number n, let e be the largest number such that n = re for

some r. Note that r is not a power of a smaller integer. We let Dk = drk, Jk = brk . It

follows that Equation (3.3.3) is equivalent to Equation (3.3.7).

Theorem 3.3.10. For any natural number n, let e, r be as described in above.

Then we have

bn =1

e

∑

l|e

µ

(e

l

)ldre.

Proof. The result follows straightforwardly from Lemma 3.3.9.

Table 2 gives the numbers of unlabeled prime graphs bn compared with the num-

bers of unlabeled connected graphs cn on no more than 12 vertices.

Table 2. Values of cn and bn, for n ≤ 12.

n cn bn

1 1 02 1 13 2 24 6 55 21 216 112 1107 853 8538 11117 111119 261080 261077

10 11716571 1171655011 1006700565 100670056512 164059830476 164059830354

3.4. Exponential Composition with a Molecular Species

In this section we introduce an operation on species that is equivalent to the

exponentiation groups in the case of molecular species. This operation turns out to

be useful for finding a functional equation relating the species of connected graphs

and the species of prime graphs.

67


Let F be a species of structures with F [∅] = ∅, let k be a positive integer, and let

A be a subgroup of Sk. Recall that an F ⊡k-structure on a finite set U is a tuple of

the form

((π1, f1), (π2, f2), . . . , (πk, fk)),

where (π1, π2, . . . , πk) is a k-rectangle on U , and each fi is an F -structure on the blocks

of πi. The group A acts on the set of F ⊡k-structures by permuting the subscripts of

πi and fi, i.e.,

α((π1, f1), . . . , (πk, fk)) = ((πα(1), fα(1)), . . . , (πα(k), fα(k))),

where α is an element of A, (πα1 , πα2 , . . . , παk) is a k-rectangle on U , and each fαi

is an F -structure on the blocks of παi. It is easy to check that this action of A on

F ⊡k-structures is natural, that is, it commutes with any bijection σ : U → V . Hence

we get a quotient species under this group action.

Definition 3.4.1. We define the exponential composition of F with the molecular

species Xk/A to be the quotient species, denoted (Xk/A)〈F 〉, under the group action

described in above. That is,

(Xk/A)〈F 〉 := F ⊡k/A.

Example 3.4.2. Figure 3.4 illustrates the group action of S2 on the set of E ⊡22 -

structures on [4], resulting in three orbits.

Let A be a subgroup of Sm, and let B be a subgroup of Sn. We have the following

formulas from Theorem 2.3.10 and Proposition 2.6.5:

Xm

A·Xn

B=

Xm+n

A > B,

68


1 2

43

1 3

2 4 4

1 3

23

1 4 1 4

32

21

4 32

Figure 3.4. There are three elements in the set E2〈E2〉[4].

Xm

A

Xn

B=

Xmn

B ≀ A,

Xm

A⊡

Xn

B=

Xmn

A × B.

Now we can add to this list one more formula representing the group action of BA

on the set N = nm stated in the following theorem.

Theorem 3.4.3. Let A and B be groups described in above, let N = nm, and let

BA be the exponentiation group of A with B, as defined in Definition 1.3.1. Then we

have

Xm

A

⟨Xn

B

⟩=

XN

BA.

Moreover, we get the cycle index of (Xm/A)〈Xn/B〉:

Z(Xm/A)〈Xn/B〉 = Z(A) ∗ Z(B),

where the expression Z(A) ∗ Z(B) denotes the image of Z(B) under the operator

obtained by substituting the operator Ir for the variables pr in Z(A).

69


Proof. Proposition 2.6.5 and the associativity of the arithmetic product give

that (Xn

B

)⊡m

=XN

Bm,

where Bm is the the product of m copies of B, acting on the set

[n] × [n] × · · · × [n]︸︷︷︸m copies

piecewise, and hence viewed as a subgroup of SN . According to Remark 2.3.4, the

set of (Xn/B)⊡m-structures on [N ] is then just the set of Bm-orbits of linear orders

on [N ].

The group A acts on these Bm-orbits of linear orders by permuting the subscripts.

This action results in the quotient species

Xm

A

⟨Xn

B

⟩=

(Xn

B

)⊡m/A =

(XN

Bm

)/A.

We observe that an A-orbit of Bm-orbits of linear orders on [N ] admits an au-

tomorphism group isomorphic to the exponentiation group BA, hence the quotient

species (XN/Bm)/A is the same as the molecular species XN/BA.

As for the cycle index of (Xm/A)〈Xn/B〉, we apply Proposition 2.3.5 and Theo-

rem 1.3.6 to get that

Z(Xm/A)〈Xn/B〉 = ZXN /BA = Z(BA) = Z(A) ∗ Z(B).

Figure 3.5 illustrates an group action of A on a set of (Xn/B)⊡m-structures.

Next, we introduce a theorem that is a generalization of Palmer and Robinson’s

Theorem 1.3.6.

70


A-orbits

B-orb

its

B-o

rbit

s

B-orbits B-orbits

B-orbits

A-orbits

A-o

rbit

sA-orb

its

A-orbits

Figure 3.5. ((Xn/B)⊡m)/A = Xnm

/BA.

Theorem 3.4.4. Let A be a subgroup of Sk, and let F be a species of structures

concentrated on the cardinality n. Then the cycle index of the species (Xk/A)〈F 〉 is

given by

(3.4.1) Z(Xk/A)〈F 〉 = Z(A) ∗ ZF .

Remark 3.4.5 (Notation and Set-up). We denote by Parn the set of partitions of

n, and by Parkn the set of k-sequences of partitions of n.

For fixed integers n, k, and N = nk, we denote by NN the species of k-dimensional

cubes, or k-cubes, on [N ], defined by

NN = E⊡kn [N ].

We also call the elements of the set (Xn)⊡k[N ] k-dimensional ordered cubes on [N ].

71


Let σ be a permutation on [k] with cycle type

c. t.(σ) = (r1, r2 . . . , rd).

Then σ acts on the F ⊡k-structures by permuting the subscripts. Let ν be a partition

of N . Let δ be a permutation of [N ] with cycle type ν. Then δ acts on the F ⊡k-

structures by transport of structures. Recall the notation introduced in Remark 1.3.8:

I(c. t.(σ); λ(1), λ(2), . . . , λ(d)) = Ir1(pλ(1)) ⊠ Ir2(pλ(2)) ⊠ · · ·⊠ Ird(pλ(d)).

We denote by RecF (σ, ν) a function on the pair (σ, ν) defined by

(3.4.2) RecF (σ, ν) :=∑ ∏d

i=1 fix F [λ(i)]

zλ(1) · · · zλ(d)

,

where the summation is over all sequences (λ(1), λ(2), . . . , λ(d)) in Pardn with

I(c. t.(σ); λ(1), λ(2), . . . , λ(d))) = pν .

We denote by fixF (σ, δ) the number of F ⊡k-structures on the set [N ] fixed by the

joint action of the pair (σ, δ).

Proof of Theorem 3.4.4. Let ν be a partition of N . It suffices to prove that

the coefficients of pν on both sides of Equation (3.4.1) are equal.

The right-hand side of Equation (3.4.1) is

Z(A) ∗ ZF =

(1

|A|

∑

σ∈A

pc.t.(σ)

)∗

(∑

λ⊢n

fix F [λ]pλ

zλ

)

=1

|A|

∑

σ∈A

Ic.t.(σ)

(∑

λ⊢n

fix F [λ]pλ

zλ

).

72


For σ ∈ A with c. t.(σ) = (r1, r2, . . . , rd), we have

Ic.t.(σ) = Ir1 · · · Ird,

and

Ic.t.(σ)

(∑

λ⊢n

fix F [λ]pλ

zλ

)

= Ir1

(∑

λ⊢n

fix F [λ]pλ

zλ

)⊠ Ir2

(∑

λ⊢n

fix F [λ]pλ

zλ

)⊠ · · · ⊠ Ird

(∑

λ⊢n

fix F [λ]pλ

zλ

).

Therefore, the coefficient of pν in the expression Z(A) ∗ ZF is

1

|A|

∑ ∏di=1 fix F [λ(i)]

zλ(1) · · · zλ(d)

=1

|A|

∑

σ∈A

RecF (σ, ν),(3.4.3)

where the summation on the left-hand side is taken over all sequences (λ(1), λ(2), . . . , λ(d))

in Pardn for some d ≥ 1 and all σ ∈ A with c. t.(σ) = (r1, r2, . . . , rd) such that

I(c. t.(σ); λ(1), λ(2), . . . , λ(d)) = pν ,

and RecF (σ, ν) on the right-hand side is as defined by (3.4.2) in Remark 3.4.5.

The left-hand side of Equation (3.4.1) is

ZF ⊡k/A =∑

ν⊢N

fixF ⊡k

A[ν]

pν

zν

.

Therefore, the coefficient of pν in the expression ZF ⊡k/A is

1

zνfix

F ⊡k

A[ν].

73


We then apply Theorem 1.1.6 to get that the number of A-orbits of F ⊡k-structures

on [N ] fixed by a permutation δ ∈ SN of cycle type ν is

fixF ⊡k

A[ν] = fix

F ⊡k

A[δ] =

1

|A|

∑

σ∈A

fixF (σ, δ),(3.4.4)

where fixF (σ, δ) is as defined in Remark 3.4.5.

Therefore, combining (3.4.4) and (3.4.3), the proof of Equation (3.4.1) is reduced

to showing that

(3.4.5) fixF (σ, δ) = zν RecF (σ, ν),

for any δ, ν and σ.

To prove (3.4.5), we start with observing that in order for an F ⊡k-structure on

[N ] of the form

((π1, f1), (π2, f2), . . . , (πk, fk))

to be fixed by the pair (σ, δ), it is necessary that (σ, δ) fixes the k-cube of the form

(π1, π2, . . . , πk) ∈ NN . This is equivalent to saying that

(3.4.6) NN [δ](π1, π2, . . . , πk) = (πσ(1), πσ(2), . . . , πσ(k)).

Suppose (3.4.6) holds for some k-cube (π1, π2, . . . , πk) ∈ NN . We let βi ∈ Sn be

the induced action of δ on the blocks of πi, for i = 1, 2, . . . , k. That is,

NN [δ](πi) = βi(πσ(i))

for all i ∈ [k].

Now we consider the simpler case when σ is a k-cycle, say, σ = (1, 2, . . . , k). Then

the action of δ sends (π1, π2, . . . , πk) to (π2, π3, . . . , π1). Let β = β1β2 · · ·βk. The

74


πσ(2)

β2 βkβ1

π1 π2 πk

πσ(k)πσ(1)

above discussion is saying that, according to Remark 1.3.3,

Ik(pc.t.(β)) = pν .

On the other hand, given a partition λ of n satisfying Ik(pλ) = pν , there are n!/zλ

permutations in Sn with cycle type λ. Let β be one of such. Then the number of

sequences (β1, β2, . . . , βk) whose product equals β is (n!)k−1, since we can choose β1 up

to βk−1 freely, and βk is thereforee determined. All such sequences (β1, β2, . . . , βk) will

satisfy Ik(pc.t.(β1···βk)) = pν , thus their action on an arbitrary k-dimensional ordered

cube, combined with the action of σ on the subscripts, would result in a permutation

on [N ] with cycle type ν. But there are N !/zν permutations with cycle type ν, and

only one of them is the δ that we started with. Considering that the k-cubes are just

Skn-orbits of the k-dimensional ordered cubes, we count the number of k-cubes that

are fixed by the pair (σ, δ) with the further condition that the product of the induced

permutations on the πi by δ has cycle type λ:

#

(β1,β2,...,βk)∈Skn

with c.t.(β1···βk)=λ

· #

k-dimensional ordered cubes

#

permutations on [N ]with cycle type ν

· #

k-dimensional ordered cubes

in each equivalence class

=[(n!)k−1 · n!/zλ] · N !

N !/zν · (n!)k=

zν

zλ

.

75


Now we try to compute how many F ⊡k-structures of the form

((π1, f1), (π2, f2), . . . , (πk, fk)),

based on a given rectangle (π1, π2, . . . , πk) that is fixed by the βi with

c. t.

(∏

i

βi

)= λ,

are fixed by the pair (σ, δ). We observe that the action of (σ, δ) determines that

fk = F [β1]f1 and fi = F [βi−1]fi−1 for i = 2, 3, . . . , k, and hence

fk = F [β1]F [β2] · · ·F [βk]fk = F [β]fk = F [λ]fk.

In other words,

fk ∈ Fix F [λ].

Hence as long as we choose an fk from Fix F [λ], then all the other fi for i < k are

determined by our choice of fk. There are fix F [λ] such choices for fk.

Therefore, in the case when σ is a k-cycle, we get that the number of F ⊡k-

structures on the set [N ] fixed by the pair (σ, δ) is

fixF (σ, δ) =∑

λ⊢nIk(pλ)=pν

fix F [λ]zν

zλ= zν RecF (σ, ν).

Now let us consider the general case when σ contains d cycles of lengths r1, r2, . . . , rd.

Let (π1, π2, . . . , πk) be a k-cube fixed by the pair (σ, δ). Again we have (3.4.6), and

we get an induced βi on the blocks of πσ−1i for each i.

We observe that the action of σ on the subscripts of the k-cube partitions the

list π1, π2, . . . , πk into d parts, of lengths r1, r2, . . . , rd, within each of which we get a

ri-cycle. We group the βi on each of the d parts and get d permutations in the group

76


Sn, whose cycle types are denoted by λ(1), λ(2), . . . , λ(d). This construction gives that

such a sequence of partitions (λ(1), λ(2), . . . , λ(d)) will be those that satisfy

I(c. t.(σ); λ(1), λ(2), . . . , λ(d)) = pν .

Therefore, the number of k-cubes fixed by (σ, δ) corresponding to such a sequence

of partitions (λ(1), λ(2), . . . , λ(d)) is

(n!)r1−1 · n!/zλ(1) · · · · · (n!)rd−1 · n!/zλ(d)

N !/zν·

N !

(n!)k

=(n!)r1+···+rd

N !

zν

zλ(1) · · · zλ(d)

=zν

zλ(1) · · · zλ(d)

.

The number of F -structures that are assigned to this k-cube (π1, π2, . . . , πk) that

will be fixed under the action of the pair (σ, δ) corresponding to the sequence of

partitions (λ(1), λ(2), . . . , λ(d)) is hence

fix F [λ(1)] · · ·fix F [λ(d)],

since, similarly to our previous discussion, within each of the d parts, we only need

to pick an F -structure that is fixed by a permutation of cycle type λ(i), and all other

F -structures are left determined.

Therefore, we get that for any pair (σ, δ),

fixF (σ, δ) = zν RecF (σ, ν),

which concludes our proof.

Remark 3.4.6. We can use the molecular decomposition to define the exponential

composition of a species F with a species H . That is, if the molecular decomposition

77


of H is given by

H =∑

M⊆HM molecular

M,

then we define H〈F 〉 by

H〈F 〉 =∑

M⊆HM molecular

M〈F 〉.

The left-linearity of the operation ∗ gives that the cycle index of H〈F 〉 is

ZH〈F 〉 = ZH ∗ ZF

=

( ∑

M⊆HM molecular

ZM

)∗ ZF

=∑

M⊆HM molecular

ZM ∗ ZF .

3.5. Exponential Composition

Let F be a species with F [∅] = ∅, and let k be a positive integer. We call the

quotient species Ek〈F 〉 = F ⊡k/Sk the exponential composition of F of order k. We

observe that an Ek〈F 〉-structure on a finite set U is a set of the form

(π1, f1), (π2, f2), . . . , (πk, fk),

where (π1, π2, . . . , πk) is a k-rectangle on U , and each fi is an F -structure on the

blocks of πi. We set E0〈F 〉 = X.

Definition 3.5.1. Let F be a species with F [∅] = F [1] = ∅. We define the expo-

nential composition of F , denoted E 〈F 〉, to be the sum of Ek〈F 〉 on all nonnegative

integers k, i.e.,

E 〈F 〉 =∑

k≥0

Ek〈F 〉.

78


Proposition 3.5.2. Let F be a species with F [∅] = ∅. Then the cycle index of

the exponential composition of F is given by

ZE 〈F 〉 = p1 +∑

k≥1

Z(Sk) ∗ ZF .

Proof. We get from Theorem 3.4.4 that for k ≥ 1,

ZEk〈F 〉 = Z(Sk) ∗ ZF .

And the rest follows from the partial linearity of the operation ∗ on symmetric func-

tions.

The exponential composition has the following properties.

Theorem 3.5.3. Let F1 and F2 be species with F1[∅] = F2[∅] = ∅, and let k be

any nonnegative integer. Then

Ek〈F1 + F2〉 =

k∑

i=0

Ei〈F1〉 ⊡ Ek−i〈F2〉,

E 〈F1 + F2〉 = E 〈F1〉 ⊡ E 〈F2〉.(3.5.1)

We observe that an (F1 +F2)⊡k-structure on a finite set U is a rectangle on U with

each partition in the rectangle enriched with either an F1 or an F2-structure. Taking

the Sk-orbits of these (F1 + F2)⊡k-structures on U means basically making every

partition of the rectangle “indistinguishable”. Hence in each Sk-orbit, all partitions

enriched with an F1-structure are grouped together to give an Sk1-orbit of F ⊡k11 -

structures, and the remaining partitions are grouped together to give an Sk2-orbit of

F ⊡k22 -structures, where k1 and k2 are nonnegative integers whose sum is equal to k.

79


Proof of Theorem 3.5.3. First, we prove that for any nonnegative integer k,

Ek〈F1 + F2〉 =k∑

i=0

Ei〈F1〉 ⊡ Ek−i〈F2〉.

The case when k = 0 is trivial. Let us consider k to be a positive integer. Let s

and t be nonnegative integers whose sum equals k. Let U be a finite set. To get an

Es〈F1〉 ⊡ Et〈F2〉-structure on U , we first take a rectangle (ρ, τ) on U , and then take

an ordered pair (a, b), where a is an Es〈F1〉-structure on the blocks of ρ, and b is an

Et〈F2〉-structure on the blocks of τ . That is,

a = (ρ1, f1), . . . , (ρs, fs), b = (τ1, g1), . . . , (τt, gt),

where (ρ1, . . . , ρs) is a rectangle on the blocks of ρ, (τ1, . . . , τt) is a rectangle on the

blocks of τ , fi is an F1-structure on the blocks of ρi, and gj is an F2-structure on the

blocks of τj .

Recall that Proposition 2.6.8 says for any nonnegative integers i, j,

N(i+j) = N

(i)⊡ N

(j).

It follows that (ρ1, . . . , ρs, τ1, . . . , τt) is a rectangle on U .

On the other hand, let x be an Ek〈F1 + F2〉-structure on U . We can write x as a

set of the form

x = (π1, f1), . . . , (πr, fr), (πr+1, gr+1), . . . , (πk, gk),

where (π1, π2, . . . , πk) is a k-rectangle on U , r is a nonnegative integer between 0 and

k, each fi is an F1-structure on πi for i = 1, . . . , r, and each gj is an F2-structure on

πj for j = r + 1, . . . , k.

80


We then write x = (x1, x2), where

x1 = (π1, f1), . . . , (πr, fr), x2 = (πr+1, gr+1), . . . , (πk, gk).

Hence running through values of s and t, we get that the set of Es〈F1〉 ⊡ Et〈F2〉-

structures on U , written in the form of the pairs (a, b) whose construction we described

in above, corresponds naturally to the set of Ek〈F1 + F2〉-structures on U .

The proof of

E 〈F1 + F2〉 = E 〈F1〉 ⊡ E 〈F2〉.

is straightforward using the properties of the arithmetic product, namely, the com-

mutativity, associativity and distributivity:

E 〈F1 + F2〉 =∑

k≥0

Ek〈F1 + F2〉

=∑

k≥0

∑

i+j=ki,j≥0

Ei〈F1〉 ⊡ Ej〈F2〉

=

(∑

i≥0

Ei〈F1〉

)⊡

(∑

j≥0

Ej〈F2〉

)

= E 〈F1〉 ⊡ E 〈F2〉.

Note that identity (3.5.1) is analogous to the identity about the composition of a

sum of species with the species of sets E :

E (F1 + F2) = E (F1) E (F2).

What is more, (3.5.1) illustrates a kind of distributivity of the exponential composi-

tion. In fact, if a species of structures F has its molecular decomposition written in

81


the form

F =∑

M⊆FM molecular

M,

then the exponential composition of F can be written as

E 〈F 〉 = ⊡M⊆F

M molecular

E 〈M〉.

3.6. Cycle Index of Prime Graphs

Now we are ready to come back to the species of prime graphs.

Lemma 3.6.1. Let P be any prime graph, and k any nonnegative integer. Then

the species associated to the k-th power of P is the exponential composition of OP of

order k. That is,

OP k = Ek〈OP 〉.

Proof. We apply Theorem 3.4.3 and get

Ek〈OP 〉 = O⊡kP /Sk =

(Xn

aut(P )

)⊡k/Sk =

Xnk

aut(P )Sk.

It follows from Remark 3.2.2 that

Ek〈OP 〉 =Xnk

aut(P k)= OP k.

We can verify Lemma 3.6.1 in an intuitive way. Note that the set of Ek〈OP 〉-

structures on a finite set U is the set of Sk-orbits of O⊡kP -structures on U , and an

element of Ek〈OP 〉[U ] of the form (π1, f1), . . . , (πk, fk) is such that (π1, π2, . . . , πk)

is a k-rectangle on U , and each fi is a graph isomorphic to P whose vertex set

equal to the blocks of πi. Such a set (π1, f1), . . . , (πk, fk) corresponds to a graph G

82


isomorphic to P k with vertex set U . More precisely, G is the Cartesian product of

the fi in which each vertex u ∈ U is of the form u = B1 ∩ B2 ∩ · · · ∩ Bk, where each

Bi is one of the blocks of πi. In this way, we get a one-to-one correspondence between

the Ek〈OP 〉-structures on U and the set of graphs isomorphic to P k with vertex set

U .

Theorem 3.6.2. The species G c of connected graphs and P of prime graphs

satisfy

Gc = E 〈P〉.

Proof. In this proof, all graphs considered are unlabeled.

The molecular decomposition of the species of prime graphs is

P =∑

P prime

OP ,

where each OP is a molecular species which is isomorphic to X l(P )/ aut(P ).

Let P1, P2, . . . be the set of unlabeled prime graphs. We have

E 〈P〉 = E 〈OP1 + OP2 + · · · 〉

= E 〈OP1〉 ⊡ E 〈OP2〉 ⊡ · · ·

= (X + OP1 + OP 21

+ · · · ) ⊡ (X + OP2 + OP 22

+ · · · ) ⊡ · · ·

=∑

i1,i2,···≥0

OP

i11

⊡ OP

i22

⊡ · · ·

=∑

i1,i2,···≥0

OP

i11 ⊡P

i22 ⊡···

=∑

C connected

OC

= Gc.

83


Now we can compute the cycle index of the species of prime graphs recursively

using Maple:

ZP =

(1

2p2

1 +1

2p2

)+

(2

3p3

1 + p1p2 +1

3p3

)

+

(35

24p4

1 +7

4p2

1p2 +2

3p1p3 +

7

8p2

2 +1

4p4

)

+

(91

15p5

1 +19

3p3

1p2 +4

3p3

1p3 + 5p1p22 + p1p4 +

2

3p2p3 +

3

5p5

)

+

(1654

45p6

1 +91

3p4

1p2 +38

9p3

1p3 + 21p21p

22 + 2p2

1p4 +8

3p1p2p3

+4

5p1p5 +

47

6p3

2 +5

2p2p4 +

11

9p2

3 +2

3p6

)+ · · · .

Figure 3.6. Unlabeled prime graphs on n vertices, n ≤ 4.

We can write down the beginning terms of the molecular decomposition of the

species P:

P = E2 + (XE2 + E3) + (E2 X2 + XE3 + X2E2 + E

22 + E4) + · · · .

Comparing Figure 3.6 with Figure 2.9, we see that there is only one unlabeled

connected graph with 4 vertices that is not prime. In fact, if we compare the first

84


several terms of ZG c , given in (2.4.6), and ZP of order no more than 6, we get that

ZG c − ZP = p1 +1

8

(p4

1 + 2p21p2 + 3 p2

2 + 2p4

)

+1

4

(p6

1 + p21p

22 + 2p3

2

)+

1

12

(p6

1 + 3p21p

22 + 4p3

2 + 2p23 + 2p6

)· · · ,

which is the cycle index of connected non-prime graphs on no more than 6 vertices,

as shown in Figure 3.7, which consist of a single vertex, a graph with 4 vertices, and

Figure 3.7. Unlabeled non-prime graphs on n vertices, n ≤ 6.

two graphs with 6 vertices.

85

CHAPTER 4

Point-Determining Graphs

4.0. Introduction

In this chapter, we examine the species of point-determining graphs (Definition 4.1.1),

which are graphs whose vertices all have distinct neighborhoods, the species of co-

point-determining graphs (Definition 4.1.2), which are graphs whose complements

are point-determining, the species of connected point-determining graphs, and the

species of bi-point-determining graphs (Definition 4.3.1), which are graphs that are

both point-determining and co-point-determining. Our goal is to find the cycle indices

of these species. First, we find a functional equation relating the species of point-

determining graphs and the well-known species of graphs (Theorem 4.1.3). At the

same time, we find a simple connection between the point-determining graphs and the

co-point-determining graphs. The connected cases are similar to the enumeration of

connected graphs as shown in Section 2.4. Furthermore, we find a functional equation

relating the species of bi-point-determining graphs and the species of graphs.

In addition, we examine the 2-sort species of 2-colored graphs (Definition 4.4.1),

which are graphs whose vertices are properly colored with white and black. We

develop ways to enumerate the 2-sort species of connected 2-colored graphs and the

2-sort species of point-determining 2-colored graphs.

86

CHAPTER 4. POINT-DETERMINING GRAPHS

4.1. Point-Determining Graphs and Co-Point-Determining Graphs

Definition 4.1.1. A point-determining graph, previously studied by Sumner [28],

also called a mating-type graph by Bull and Pease [4] and Read [23], is a graph G

in which any two distinct vertices have distinct neighborhoods, i.e., if v1 6= v2, then

N(v1) 6= N(v2), where N(v) = w : wv is an edge of G is the set of vertices adjacent

to v.

Note that the neighborhood of an isolated vertex is the empty set. Thus in a

point-determining graph, there is at most one isolated vertex.

Definition 4.1.2. A graph is called co-point-determining if its complement is point-

determining.

In a co-point-determining graph, two non-adjacent distinct vertices have distinct

augmented neighborhoods. That is, if v1 and v2 are distinct vertices in this graph,

then N(v1) ∪ v1 6= N(v2) ∪ v2.

We denote the species of point-determining graphs by P. Thus in the species

language, a point-determining graph is a P-structure. We denote by Q the species

of co-point-determining graphs. We set the number of point-determining graphs and

the number of co-point-determining graphs on an empty set of vertices both to be

one.

We observe that there is a natural transformation α that produces for every finite

set U a bijection between P[U ] and Q[U ], which sends each point-determining graph

with vertex set U to its complement, which is a co-point-determining graph with

vertex set U . Furthermore, the following diagram commutes for any finite sets U , V

and any bijection σ : U → V :

87


P[U ]P[σ]−−−→ P[V ]

α

yyα

Q[U ]Q[σ]

−−−→ Q[V ]Hence, according to Remark 2.1.7, the species P and Q are isomorphic, and we

do not make distinctions between them during calculations.

There are four point-determining graphs and and four co-point-determining graphs

labeled on [3], as shown in Figure 4.1, whose first row consists of the point-determining

graphs and whose second row consists of the co-point-determining graphs.

1

32

1

32

1

32

1

32

1

32

1

32

1

32

1

32

Figure 4.1. Labeled point-determining graphs and co-point-determining graphs with vertex set [3].

The following theorem gives a starting point for counting point-determining graphs.

Theorem 4.1.3. Let G be the species of graphs, P the species of point-determining

graphs, and E+ the species of nonempty sets. Then we have

(4.1.1) G = P(E+).

Proof. Given any graph G with vertex set V , we define an equivalence relation

on V by setting two elements v and w of V to be equivalent to each other if they

88


have the same neighborhoods. This gives a partition of V into m equivalence classes

π = V1, V2, . . . , Vm.

We then associated to the graph G the ordered pair (π, G′), where G′ is the graph

whose vertices are the blocks of π in which two vertices Vi and Vj in G′ are adjacent if

and only if there is an edge in G connecting a vertex in Vi with one in Vj . An example

of such a transformation from a graph G to a point-determining graph G′ is illustrated

in Figure 4.2. It is straightforward to see that the graph G′ is point-determining, since

no two vertices of G′ have the same neighborhoods.

4

3

1

9 2

6

5

74

8

3

1

9 2

6

5

7

8

Figure 4.2. A transformation from a graph G with vertex set [11]to a point-determining graph G′ with vertex set1, 9, 3, 8, 4, 7, 6, 2, 5.

On the other hand, we can construct a graph on n vertices uniquely (up to isomor-

phism) from an ordered pair consisting of a point-determining graph on m vertices

and a partition of n vertices into m blocks by reversing the procedure described in

above. This bijection leads to the species equivalence relation (4.1.1).

89


Using the same method we can prove that

(4.1.2) G (X) = Q(K+),

where K+ is the species of complete graphs with non-empty vertex sets.

Recall Γ, the compositional inverse of E+ as defined in Section 2.4. Equations (4.1.1)

and (4.1.2) can be rewritten as

P = Q = G (Γ).

This identity gives rise to several identities that can be used to compute the associated

series of P:

P(x) = Q(x) = G (log(1 + x)),(4.1.3)

P(x) = Q(x) = ZG (x − x2, x2 − x4, . . . ),(4.1.4)

ZP = ZQ = ZG

(∑

k≥1

µ(k)

klog(1 + pk),

∑

k≥1

µ(k)

klog(1 + p2k), . . .

).

Read [23] derived formulas (4.1.3) and (4.1.4), and pointed out that identity (4.1.3)

gives an explicit expression for the numbers of labeled point-determining graphs with

k vertices pk:

pk =∑

n≥0

2(n2 ) s(n, k),

where the s(n, k) denote Stirling numbers of the first kind [29].

Using Maple, we can write down the first several terms of the associated series of

P:

P(x) = 1 +x

1!+

x2

2!+ 4

x3

3!+ 32

x4

4!+ 588

x5

5!+ 21476

x6

6!+ 1551368

x7

7!+ · · · ,

P(x) = 1 + x + x2 + 2 x3 + 5 x4 + 16 x5 + 78 x6 + 588 x7 + 8047 x8 + 205914 x9 + · · · ,

90


ZP = 1 + p1 +

(1

2p2

1 +1

2p2

)+

(1

3p3 + p1p2 +

2

3p3

1 +3

2p2

1p2

)

+

(1

2p4 +

4

3p4

1 + p22 +

2

3p1p3

)

+

(p2

1p3 +49

10p5

1 +11

3p3

1p2 +1

3p2p3 +

3

5p5 + p1p4 +

9

2p1p

22

)+ · · · .

Figure 4.3. Unlabeled point-determining graphs on n vertices, n ≤ 4.

In this way, the molecular decomposition of P takes the form

P = X + E2 +

(XE2 + E3

)+

(X2

E2 + XE3 + E2 E2 + X2E2 + E4

)+ · · · .

Let pn, cn and gn be numbers of labeled point-determining graphs, connected

graphs and graphs on n vertices. We have the following table:

4.2. Connected Point-Determining Graphs and Connected

Co-Point-Determining Graphs

A point-determining or a co-point-determining graph is called connected if the

graph itself is connected. We denote by Pc the species of connected point-determining

graphs, and by Qc the species of connected co-point-determining graphs.

91


Table 1. Numbers of labeled point-determining graphs, connectedgraphs and graphs on n vertices, n ≤ 10.

n pn cn gn

1 1 1 12 1 1 23 4 4 84 32 38 645 588 728 10246 21476 26704 327687 1551368 1866256 20971528 218608712 251548592 2684354569 60071657408 66296291072 68719476736

10 32307552561088 34496488594816 35184372088832

Theorem 4.2.1. Let P be the species of point-determining graphs, and let Pc be

the species of connected point-determining graphs. We have

P = (1 + X) E (Pc − X).

Proof. Since a point-determining graph can have at most one isolated vertex,

and the rest of its connected components are connected point-determining graphs

with at least two vertices, we have

P = E (Pc − X) + XE (Pc − X).

Theorem 4.2.2. For the species Q and Qc, we have

Q = E (Qc).

Proof. The result follows straightforwardly from the observation that a graph

is co-point-determining if and only if all its connected components are.

92


Since P and Q are isomorphic to each other, we get

(4.2.1) P = (1 + X) E (Pc − X) = E (Qc).

Recall the virtual species Γ, which is the compositional inverse of E+ introduced

in Section 2.4, we have

Qc = Γ(P+),

where P+ is the species of point-determining graphs with nonempty vertex sets.

Hence we get to compute the associated series of Qc:

Qc(x) = log(P(x)),

Qc(x) =∑

k≥1

µ(k)

klog(P(xk)),

ZQc =∑

k≥1

µ(k)

klog(ZP pk − pk + 1).

We write down the first several terms of the associated series of Qc using Maple:

Qc(x) = x + 3

x3

3!+ 19

x4

4!+ 462

x5

5!+ 18268

x6

6!+ 1410394

x7

7!+ 206677954

x8

8!+ . . . ,

Qc(x) = x + x3 + 3x4 + 11x5 + 61x6 + 507x7 + 7442x8 + 197772x9 + 9808209x10 + · · · ,

ZQc = p1 +

(1

2p1p2 +

1

2p3

1

)+

(19

24p4

1 +3

4p2

1p2 +1

3p1p3 +

7

8p2

2 +1

4p4

)

+

(7

3p3

1p2 +1

6p2p3 +

77

20p5

1 +13

4p1p

22 +

1

2p2

1p3 +2

5p5 +

1

2p1p4

)+ · · · .

Equation (4.2.1) can be rewritten as

(4.2.2) P = E (Γ + Pc − X) = E (Qc).

93


As a general fact, if E (F1) = E (F2) for any species F1 and F2, then

E+(F1) = E+(F2).

It follows that Γ E+(F1) = Γ E+(F2). Since Γ E+ = X, we have F1 = F2.

Therefore, Equation (4.2.2) gives

Γ + Pc − X = Q

c,

or, equivalently,

Γ = Qc − P

c≥2,

which gives an explicit expression for the virtual species Γ as the difference of two

species.

We also get the functional equations relating the associated series of Pc and those

of Qc:

Pc(x) = Q

c(x) + x − log(1 + x),

Pc(x) = Qc(x) + x − (x − x2),

ZPc = ZQc + p1 −∑

k≥1

µ(k)

klog(1 + pk).

Through computation, we get the first several terms of the associated series of

Pc:

Pc(x) = x +

x2

2!+

x3

3!+ 25

x4

4!+ 438

x5

5!+ 18388

x6

6!+ 1409674

x7

7!+ 206682994

x8

8!+ . . . ,

Pc(x) = x + x2 + x3 + 3x4 + 11x5 + 61x6 + 507x7 + 7442x8 + 197772x9 + · · · ,

ZPc = p1 +

(1

2p2

1 +1

2p2

)+

(1

6p3

1 +1

3p3 +

1

2p1p2

)

94


+

(25

24p4

1 +5

8p2

2 +3

4p2

1p2 +1

3p1p3 +

1

4p4 +

1

2p2

1p3 +13

4p1p

22

)

+

(7

3p3

1p2 +3

5p5 +

73

20p5

1 +1

6p2p3 +

1

2p1p4

)+ · · · .

Let pcn and qc

n be the numbers of labeled connected point-determining graphs and

connected co-point-determining graphs, respectively, so that

Pc(x) =

∑

n≥1

pcn

xn

n!, Q

c(x) =∑

n≥1

qcn

xn

n!.

We get from the above species equivalence an identity relating pcn and qc

n:

(4.2.3) pcn + (−1)n−1(n − 1)! = qc

n, for n ≥ 2.

Figure 4.4 shows the unlabeled connected point-determining graphs and unlabeled

connected co-point-determining graphs with 4 vertices.

p1 p2 p3 q1 q2 q3

Figure 4.4. Unlabeled connected point-determining graphs and un-labeled connected co-point-determining graphs on 4 vertices.

We can get the number of labeled connected point-determining graphs and co-

point-determining graphs by calculating the number of labeled graphs isomorphic to

each pi and qi:

pc4 =

(24

| aut(p1)|+

24

| aut(p2)|+

24

| aut(p3)|

)= 12 + 12 + 1 = 25,

qc4 =

(24

| aut(p1)|+

24

| aut(p2)|+

24

| aut(p3)|

)= 12 + 3 + 4 = 19.

95


A combinatorial proof of Equation (4.2.3) would be desirable.

Table 2. Values of pcn and qc

n , for n ≤ 10.

n pcn qc

n

1 1 12 1 03 1 34 25 195 438 4626 18388 182687 1409674 14103948 206682994 2066779549 58152537184 58152577504

10 31715884061624 31715883698744

4.3. Bi-Point-Determining Graphs

Definition 4.3.1. A graph is called bi-point-determining if it is both point-determining

and co-point-determining.

The enumeration of the bi-point-determining graphs is carried out through the

structures called phylogenetic trees.

Definition 4.3.2. A phylogenetic tree is a rooted tree with labeled leaves and unla-

beled internal vertices in which no vertex has exactly one child.

24

5

1

3

6

Figure 4.5. A phylogenetic tree on [6].

96


Theorem 4.3.3. Let R be the species of bi-point-determining graphs. Let G and

S be the species of graphs and phylogenetic trees respectively. Then we have

G = R(2S − X).

The enumeration of phylogenetic trees was studied by Carlitz and Riordan [5],

Foulds and Robinson [7], Lomnicki [14], and others.

Lemma 4.3.4. Let S be the species of phylogenetic trees, and let E≥2 be the species

of sets with no less than two elements. We have

(4.3.1) S = X + E≥2(S ).

Proof. A phylogenetic tree is either a singleton vertex, which contributes to the

term X on the right-hand side of Equation (4.3.1), or is a phylogenetic tree with at

least two leaves, in which case we can separate the root and the rest of the tree and

get a set of at least two subtrees each of which is again a phylogenetic tree, which, as

illustrated in Figure 4.6, is an E≥2 S -structure.

Figure 4.6. A phylogenetic tree with at least two leaves is a set ofat least two subtrees each of which is a phylogenetic tree.

97


Equation (4.3.1) enables us to compute the first several terms of the associated

series of S using Maple:

S (x) = x +x2

2!+ 4

x3

3!+ 26

x4

4!+ 236

x5

5!+ 2752

x6

6!+ 39208

x7

7!+ 660032

x8

8!+ · · · ,

S (x) = x + x2 + 2x3 + 5x4 + 12x5 + 33x6 + 90x7 + 261x8 + 766x9 + · · · ,

ZS = p1 +

(1

2p2

1 +1

2p2

)+

(2

3p3

1 + p1p2 +1

3p3

)

+

(13

12p4

1 + p21p2 +

2

3p1p3 +

3

4p2

2 +1

2p4

)

+

(59

30p5

1 +4

3p2

1p3 + 2 +5

2p1p

22 +

13

3p3

1p2 + p1p4 +2

3p2p3 +

1

5p5

)

+

(172

45p6

1 +2

5p1p5 +

59

6p4

1p2 +26

9p3

1p3 +15

2p2

1p22 + 2p2

1p4 +8

3p1p2p3

+3

2p2p4 +

1

2p6

)+ · · · .

Figure 4.7 shows the unlabeled phylogenetic trees with no more than 5 vertices.

Figure 4.7. Unlabeled phylogenetic trees on n vertices, n ≤ 5.

98


Therefore, the molecular decomposition of S is

S = X + E2 +

(XE2 + E3

)+

(X2

E2 + E2 E2 + E22 + XE3 + E4

)

+

(X3

E2 + X2E3 + 3XE

22 + 2XE2 E2 + XE4 + 3E2E3 + E5

)+ · · · .

It is of interest to observe that different trees might have the same species ex-

pression. For example, the three unlabeled phylogenetic trees with 5 vertices listed

in Figure 4.8 correspond to the same species XE 22 .

Figure 4.8. Unlabeled phylogenetic trees corresponding to thespecies XE 2

2 .

Definition 4.3.5. We define an alternating phylogenetic tree to be either a single

vertex, or a phylogenetic tree with more than one labeled vertex whose internal ver-

tices are colored black or white, where no two adjacent vertices are colored the same

way.

We denote by S ∗ the species of alternating phylogenetic trees. The structure of

alternating phylogenetic trees S ∗ was studied by Stanley [27, p. 89], MacMahon [16,

17], and Riordan and Shannon [24] under other names such as yoke-chains and series-

parallel networks.

Note that any phylogenetic tree with more than one labeled vertex gives rise to two

alternating phylogenetic trees, since the coloring of the internal vertices is uniquely

99


8

3

7

69

5

1

2

4

Figure 4.9. An alternating phylogenetic tree labeled on the set [9],where the root is colored black.

determined by the coloring of the root, which has two choices. Therefore,

S∗ = 2S − X.

Proof of Theorem 4.3.3. In this proof, we will show how to transform an

arbitrary graph into a bi-point-determining graphs. This transformation, with all in-

formation preserved using the structure of alternating phylogenetic trees, is reversible,

and hence gives a bijection from the set of graphs with vertex set U to the set of triples

of the form (π, ϕ, γ) where π is a partition of U , ϕ is a bi-point-determining on the

blocks of π, and γ is a set of alternating phylogenetic trees each of which is labeled

on a block of π. Such a bijection leads to the functional equation

G = R S∗.

Let (g, h) be an ordered pair that satisfies the following conditions

C1 g is a graph with vertex set u = u1, u2, . . . , where each ui is a set, and the

ui are disjoint;

C2] h is a set h = t1, t2, . . ., where each ti is an alternating phylogenetic tree

labeled on the vertex set ui. Such pairs (g, h) are illustrated in Figures 4.10 and 4.11.

100


t3t2

g with V (g) = u1, u2, u3, u4

t4t1

h = t1, t2, t3, t4

u1

u4u2 u3

Figure 4.10. A pair (g, h) that satisfies conditions C1 and C2.

Next, we define two operations OQ and OP on pairs (g, h) such that each operation

sends (g, h) to a new ordered pair (g′, h′).

First, the operation OQ sends (g, h) to a new pair (g′, h′), where g′ is a graph and

h′ is a set of alternating phylogenetic trees. More precisely, we start by defining an

equivalence relation on the vertex set of g such that two vertices are equivalent if they

have the same augmented neighborhoods. We denote by

E = e1, e2, . . . , em

the set of equivalence classes of V (g), where each ei is a set of vertices of g, i.e.,

ei = ui,1, ui,2, . . . .

We let w1, w2, . . . be such that

wi = ui,1 ∪ ui,12 ∪ · · · .

We now let the wi be the vertices of g′, and two vertices wi and wj are adjacent in g′

if a vertex in the equivalence class ei is adjacent to a vertex in the equivalence class

101


ej in the graph g. Note that

w1 ∪ w2 ∪ · · · = u1 ∪ u2 ∪ · · · .

We construct a set

h′ = t′1, t′2, . . .

by letting t′i be the alternating phylogenetic tree whose root is colored white and whose

children are the trees ti,1, ti,2, · · · ∈ h labeled by the sets ui,1, ui,2, · · · ∈ V (g). Note that

t′i could fail to be an alternating phylogenetic tree, because some of the alternating

phylogenetic trees corresponding to the vertices ui,1, ui,2, . . . of g have white roots. In

this case, in order to make t′i have alternating colors on the internal vertices, we need

to modify our construction of t′i by taking each white-rooted alternating phylogenetic

tree, ti,k, and attaching the children of the root of ti,k directly to the root of t′i. We

get a new pair (g′, h′), where g′ is a graph on two vertices w1 and w2 with

w1 = u1, w2 = u2 ∪ u3 ∪ u4,

and h′ is a set of two alternating phylogenetic trees, one on the set w1, the other on

the set w2. Note that the alternating phylogenetic tree on u2 has a white root, so we

attach the children of this tree to the white root in the alternating phylogenetic tree

on w2.

Second, the operation OP sends the ordered pair (g, h) to a new ordered pair

(g′, h′) in a way similar to the operation OQ, except that now the equivalence classes

on the vertices of g are taken over vertices with the same neighborhoods in g, and

that the new root we attach to a set of alternating phylogenetic trees is colored black

instead of white. Again, we modify our construction as in OQ to ensure that we get

new alternating phylogenetic trees without violating the rule of alternating colors.

102


4 1

1

1

3

2

4

2

1

2,3

2 33 4

2 3

1 4

3

4

41 2

1

1

2

2,3,4

3

4

a)

(g′, h′)

(g, h) (g, h)

(g′, h′)

b)

Figure 4.11. a) An OQ application. b) An OP application.

Figure 4.12 illustrates the operation OP on a pair (g, h) in Figure 4.10, where

V (g) = u1, u2, u3, u4,

h is a set of alternating phylogenetic trees labeled on the set ui for i = 1, 2, 3, 4.

h′ = t′1, t′2

t′1

t′2

g′ with V (g′) = w1, w2

t′2

t′1

w1

w2

Figure 4.12. The operation OP applied to a pair (g, h) in Figure 4.10requires a modification on the construction of alternating phylogenetictree t′2.

103


It is straightforward to check that the new pair (g′, h′) we get under either the

application of OQ or OP satisfies conditions C1 and C2, and that the union of the

vertices of g is equal to the union of the vertices of g′. We further observe that the

operation OQ sends a graph g to a co-point-determining graph g′, since in g′, no two

vertices have the same augmented neighborhoods. Similarly, the operation OP sends

a graph g to a point-determining graph g′. That is why OP and OQ should be applied

alternatingly, since OP has no effect on a point-determining graph, and OQ has no

effect on a co-point-determining graph.

Now let G be a graph with vertex set

U = v1, v2, . . . .

We relabel G with the set

U ′ = U1, U2, . . . ,

where each Ui = vi is a singleton set, viewed as the label of a vertex of G. We

assign to G a set

H = T1, T2, . . . ,

where each Ti is a single vertex vi. Then (G, H) is an ordered pair satisfying conditions

C1 and C2.

We keep applying the operations OQ and OP alternatingly on the pair (G, H),

until neither operation has any effect, that is, when we reach an ordered pair (Gk, Hk)

in which Gk is a bi-point-determining graph. Illustrated in Figure 4.13 is a sequence

of alternating operations OQ and OP , starting with OP , sending a pair (G, H) to a

pair (Gk, Hk), where Gk is a bi-point-determining graph on a single vertex, and Hk is

a set consists of a single alternating phylogenetic tree whose vertex set is the vertex

of Gk.

104


2,3

5

1

4

4

1

2

2

1

3

54

4,5

1

1 2 3 4 5

3

1,2,3,4,5

4,5

2

5

1

4

3

3

3

2

2

321 4 52,3,4,5

1

1 2 3

1

4

1

5

2,3,4,5

5

Figure 4.13. Alternating applications of OP and OQ send a pair(G, H) to a pair (Gk, Hk).

The above discussion shows that we get from any graph G, we get a pair (Gk, Hk)

satisfies conditions C1 and C2. To be more precise, writing

V (Gk) = V1, V2, . . . , Hk = T1, T2, . . . ,

we get a triple of the form (π, ϕ, γ), where

i) π = V1, V2, . . . is a partition of U ;

ii) ϕ = Gk is a bi-point-determining graph with vertex set π.

iii) γ = T1, T2, . . . , where for each i, Ti is an alternating phylogenetic tree with

vertex set Vi.

On the other hand, since all information is preserved using the alternating phylo-

genetic trees, the above procedure is reversible. Roughly speaking, on each step, the

adjacency outside the equivalence classes are always preserved, while the adjacency

between vertices with the same augmented neighborhoods are transformed into them

105


located in the same alternating phylogenetic tree with a white common ancestor,

where the common ancestor of two vertices a and b in a phylogenetic tree is defined

to be such that if we take the unique shortest path from a to b, say, w0w1 · · ·wl, with

w0 = a and wl = b, then the common ancestor of a and b is the unique wi for which

both wi−1 and wi+1 are children of wi.

More precisely, suppose we are given a triple (π, ϕ, γ), where π = V1, V2, . . . is

a partition of U , i.e., ∪iVi = U , ϕ is a bi-point-determining graph on the blocks of π,

and γ is a set S1, S2, . . . in which each Si is an alternating phylogenetic tree labeled

on the set Vi. Then there is a unique graph G with vertex set U constructed in the

way such that for any v1, v2 ∈ V (G), v1, v2 is an edge of G if and only if exactly

one of the following two conditions is satisfied:

• v1, v2 ∈ Vi for some i, hence v1 and v2 are labels of vertices of Si. Then we

require the common ancestor of those vertices labeled v1 and v2 in Si to be

colored white. The common ancestor of two vertices a and b in a phylogenetic

tree is defined to be such that if we take the unique shortest path from a to

b, say, w0w1 · · ·wl, with w0 = a and wl = b, then the common ancestor of a

and b is the unique wi for which both wi−1 and wi+1 are children of wi.

• v1 ∈ Vi, v2 ∈ Vj and i 6= j. Then we require Vi and Vj to be adjacent to each

other in the graph ϕ.

The above can be concluded with a species equivalence

G = R S∗,

which gives

S = R (2S − X).

106


86

2

4

1

8

6 3

3

4

7

1 527

5

V1 V2

V4V3

V2

V3 V4

V1

Figure 4.14. Construct a graph G from a given triple (π, ϕ, γ).

We get from Theorem 4.3.3 a functional equation expressing the species R in

terms of the species G .

Corollary 4.3.6. Let G , R be the species of graphs and the species of bi-point-

determining graphs. Let Γ be the virtual species as defined in Section 2.4. Then we

have

(4.3.2) R = G (2Γ − X).

Proof. First, we observe that since S0 = 0 and S1 = X, the same is true for

S ∗: S ∗0 = 0 and S ∗

1 = X. Proposition 19 of [2, p. 130] says that there exists a

unique virtual species S ∗〈−1〉 such that

S∗ S

∗〈−1〉 = S∗〈−1〉 S

∗ = X.

Since Theorem 4.3.3 gives that

R = G (S ∗〈−1〉),

107


it suffices to show that the compositional inverse of S ∗ is 2Γ − X.

But 2S − S ∗ = X implies that

S =X + S ∗

2.

On the other hand, Equation (4.3.1) implies 2S − X = E (S ) − 1. So

S∗ = E (S ) − 1,

and

(4.3.3) S∗ + 1 = E (S ) = E

(X + S ∗

2

).

But from

E Γ(X) = X + 1,

we get

E Γ S∗ = X S

∗ + 1,

so

E (Γ(S ∗)) = S∗ + 1 = E

(X + S ∗

2

)

by Equation (4.3.3). So

Γ(S ∗) =X + S ∗

2,

which gives that

X = 2Γ(S ∗) − S∗ = (2Γ − X) S

∗.

108


Equation (4.3.2) gives rise to several identities for computing the associated series

of R:

R(x) = G (2 log(1 + x) − x),

R(x) = ZG (x − 2x2, x2 − 2x4, . . . ),

ZR = ZG

(2∑

k≥1

µk

klog(1 + pk) − p1, 2

∑

k≥1

µk

klog(1 + p2k) − p2, . . .

).

Using Maple, we get the following:

R(x) =x

1!+ 12

x4

4!+ 312

x5

5!+ 13824

x6

6!+ 1147488

x7

7!+ 178672128

x8

8!+ · · · ,

R(x) = x + x4 + 6x5 + 36x6 + 324x7 + 5280x8 + 156088x9 + 8415760x10 + · · · ,

(4.3.4)

ZR = p1 +

(1

2p4

1 +1

2p2

2

)+

(13

5p5

1 + 3p1p22 +

2

5p5

)

+

(96

5p6

1 + 11p21p

22 +

4

5p1p5 +

11

3p3

2 + p23 +

1

3p6

)+ · · · .

It is interesting to observe from Equation (4.3.4) that there is no bi-point-determining

graphs on 2 or 3 vertices. The unlabeled bi-point-determining graphs on 4 or 5 vertices

are shown in Figure 4.15.

Figure 4.15. Unlabeled bi-point-determining graphs on n vertices,n = 4, 5.

109


Therefore, we get the molecular decomposition of R:

R = X + E2 X2 + [5X(E2 X2) + D5] + · · · ,

where D5 = X5/D5 is the molecular species of regular pentagons with cycle index

ZD5 = Z(D5) =1

10(p5

1 + 5p1p22 + 4p5).

Listed in Table 3 are numbers of labeled and unlabeled bi-point-determining

graphs, denoted by rln and ru

n.

Table 3. Numbers of labeled and unlabeled bi-point-determininggraphs with no more than 15 vertices.

n rln ru

n

1 1 12 0 03 0 04 12 15 312 66 13824 367 1147488 3248 178672128 52809 52666091712 156088

10 29715982846848 841576011 32452221242518272 82079360012 69259424722321036032 14506388148013 291060255757818125657088 4679331014916814 2421848956937579216663491584 2779080872680384015 40050322614433939228627991906304 30630967638347575456

We denote by Rc the species of connected bi-point-determining graphs.

Theorem 4.3.7. For species R, Rc, and E+, we have

R = X + (1 + X) E+(Rc − X).

110


Proof. Since a graph with a singleton vertex is an Rc-structure, we denote by

Rc − X the species of connected bi-point-determining graphs with more than one

vertex.

Let R be a bi-point-determining graph. Then R is both point-determining and

co-point-determining. In particular, the decomposition of R into a set of its con-

nected components is similar with that of a point-determining graph in that there

is at most one connected component consisting of a singleton vertex. Hence such a

decomposition can result in one of the following three situations:

i) R is a singleton vertex, in which case we get an X-structure.

ii) R has more than one vertex, and none of its connected components is a singleton

vertex, in which case we get an E+(Rc − X).

iii) R has more than one vertex, and one of its connected components is a singleton

vertex, in which case we get an X · E+(Rc − X) species.

Therefore, we get the desired result.

We can compute the beginning terms of the associated series of Rc using Maple:

Rc(x) =

x

1!+ 12

x4

4!+ 252

x5

5!+ 12312

x6

6!+ 1061304

x7

7!+ 170176656

x8

8!

+ 51134075424x9

9!+ · · ·

Rc(x) = x + x4 + 5x5 + 31x6 + 293x7 + 4986x8 + · · · ,

ZRc = p1 +

(1

2p4

1 +1

2p2

2

)+

(21

10p5

1 +5

2p1p

22 +

2

5p5

)

+

(17

10p6

1 +17

2p2

1p22 +

2

5p1p5 +

11

3p3

2 + p23

1

3p6

)+ · · · .

The calculation agrees with the observation from Figure 4.15 that the only un-

labeled bi-point-determining graph on four vertices is connected, and among the six

111


unlabeled bi-point-determining graph on five vertices there is only one of them that

is not connected.

4.4. 2-Colored Graphs and Connected 2-colored Graphs

Definition 4.4.1. A proper coloring of a graph is an assignment of colors to the

vertices of the graph where no two adjacent vertices are assigned the same color. A

2-colored graph, also called a bi-colored graph by Harary [11, p. 93] and Hanlon [9],

is a graph in which all vertices are properly 2-colored.

For simplicity, we call the two colors in a 2-colored graph white and black. We

denote by G (X, Y ) the 2-sort species defined such that for a two-set U = (V, W ),

G [U ] is a 2-colored graph in which the vertices colored white are elements of V and

the vertices colored black are elements of W .

G (X, Y )X

Y

Y

X

Y

Figure 4.16. The 2-sort species of 2-colored graphs G (X, Y ).

Proposition 4.4.2. Let G (X, Y ) be the 2-sort species of 2-colored graphs. Then

the exponential generating series of G (X, Y ) is given by

G (x, y) =∞∑

m,n=0

2mn xm

m!

yn

n!.

112


Proof. In a 2-colored graph, each edge must connect one vertex of sort X and

one vertex of sort Y . Hence there are 2mn labeled 2-colored graphs with m white

vertices and n black vertices.

Theorem 4.4.3. Let G (X, Y ) be the 2-sort species of 2-colored graphs. Then the

cycle index of G (X, Y ) is given by

ZG (X,Y ) =∑

m,n≥0

( ∑

λ⊢m, µ⊢n

2∑

i,j gcd(λi, µj)pλ[x]

zλ

pµ[y]

zµ

).

Proof. Let (λ, µ) be an ordered pair of partitions, and let (σ, π) be an ordered

pair of permutations with σ having cycle type λ and π having cycle type µ. Let

fix (σ, π) = fix G [λ, µ] be the number of 2-colored graphs fixed by (σ, π).

To start with, we consider the simpler case when σ is a k-cycle and π is an l-cycle.

Let Kk,l denote the complete bipartite graph on [k, l], and let E(Kk,l) be its edge set.

Then |E(Kk,l)| = kl. Without loss of generality, we let the labeling of left-hand side

vertices of Kk,l be 1, 2, . . . , k, and the labeling of right-hand side vertices of Kk,l be

1′, 2′, . . . , l′. Then each edge of Kk,l is represented by an ordered pair (i, j′), for some

i ∈ [k] and j ∈ [l]. The pair of permutations (σ, π) acts on the set E(Kk,l) by letting σ

act on the set 1, 2, . . . , k and π act on the set 1′, 2′, . . . , l′. This action partitions

the kl edges of Kk,l into orbits A1, A2, . . . . We observe that there are lcm(k, l)

edges in each of the orbits, since all edges of the form (ir, j′r), where ir = σr(i) and

jr = πr(j) for some r = 1, 2 . . . , lcm(k, l)− 1, are in the same orbit as the edge (i, j ′),

and hence this action of (σ, π) on the set E(Kk,l) results in (kl)/lcm(k, l) = gcd(k, l)

orbits. Note that each 2-colored graph with vertex set [k, l] can be identified with a

subset of E(Kk,l). If a subset S of E(Kk,l) is fixed by the pair of permutations (σ, π),

then whenever an edge (i, j′) is in S, all edges in the same orbit as (i, j′) under the

action of (σ, π) on E(Kk,l) is in S as well. This means that the number of 2-colored

113


graphs fixed by the pair of permutations (σ, π) is the same as the number of subsets

of A1, A2, . . . , Agcd(k,l). Therefore,

fix (σ, π) = 2gcd(k,l).

For the general case, we write λ = (λ1, λ2, . . . ) and µ = (µ1, µ2, . . . ). It is straight-

forward to see that each ordered pair (λi, µj), for some integers i and j, gives rise to

a factor 2gcd λi,µj in the number fix (σ, π), and hence

fix G [λ, µ] =∏

i,j

2 gcd(λi, µj) = 2∑

i,j gcd(λi, µj).

Remark 4.4.4. The above argument also gives a way to count 2-colored graphs by

the number of edges. Let bm,n(x) be the ordinary generating function for 2-colored

graphs, in which m vertices are colored white and n vertices colored black, by the

number number of edges. We get the following expression for bm,n(x), which agrees

with the result of Harary and Palmer [11, p. 95]:

bm,n(x) =∑

λ⊢m, µ⊢n

1

zλzµ

m,n∏

k,l=1

(1 + xlcm(k, l)

)ck(λ)cl(µ) gcd(k, l)

,

where ci(λ) denotes the number of parts in λ with length i. For example, the coeffi-

cient of x4 in b2,3(x) is 3, as shown in Figure 4.17.

Figure 4.17. There are 3 unlabeled 2-colored graphs with 4 edgesand 5 vertices, 2 colored white, 3 colored black.

114


Theorem 4.4.3 enables us to calculate the associated series of G [X, Y ]:

G (x, y) = 1 +x

1!+

x2

2!+

x3

3!+

y

1!+

y2

2!+

y3

3!+ 2

x

1!

y

1!+ 4

x2

2!

y

1!+ 4

x

1!

y2

2!+ · · · ,

G (x, y) = 1 + x + y + 2xy + x2 + y2 + 3x2y + 3xy2 + x3 + y3 + · · · ,

ZG (X,Y ) = 1 + (p1[x] + p1[y]) +

(1

2p2

1[x] +1

2p2[x] + 2p1[x]p1[y] +

1

2p2[y] +

1

2p2

1[y]

)

+

(1

6p3

1[x] +1

2p1[x]p2[x] +

1

3p3[x] + p2[x]p1[y] + 2p2

1[x]p1[y]

+2p1[x]p21[y] + p1[x]p2[y] + +

1

3p3[y] +

1

2p1[y]p2[y] +

1

6p3

1[y]

)+ · · · .

Figure 4.18 shows the unlabeled 2-colored graphs on at most 4 vertices, where

each vertex is either a white vertex, a black vertex, or a vertex that can be colored

in both colors. For example, there are two unlabeled 2-colored graph on one vertex,

one color black the other colored white.

Figure 4.18. Unlabeled 2-colored graphs with no more than 4 vertices.

A 2-colored graph is called connected if the underlying graph is connected.

115


Theorem 4.4.5. Let G c(X, Y ) be the species of connected 2-colored graphs. Then

(4.4.1) G (X, Y ) = E (G c(X, Y )).

Proof. The canonical decomposition of a graph into connected components ap-

plies to 2-colored graphs.

Equation (4.4.1) is equivalent to

Gc(X, Y ) = Γ(G+(X, Y )),

which enables us to compute the associated series of G c[X, Y ] using Maple:

Gc(x, y) =

x

1!+

y

1!+

x

1!

y

1!+

x2

2!

y

1!+

x

1!

y2

2!+

x3

3!

y

1!+ 5

x2

2!

y2

2!+

x

1!

y3

3!+ · · · ,

G c(x, y) = x + y + xy + xy2 + x2y + x3y + xy3 + 2x2y2 + x4y + 4x3y2 + 4x2y3

+ xy4 + · · · ,

ZG c(X,Y ) = (p1[x] + p1[y]) + p1[x]p1[y] +

(1

2p2

1[x]p1[y] +1

2p1[x]p2

1[y] +1

2p2[x]p1[y]

+1

2p1[x]p2[y]

)+ · · · .

Figure 4.19. Unlabeled connected 2-colored graphs with no morethan 4 vertices.

116


4.5. Point-Determining 2-Colored Graphs

Definition 4.5.1. A 2-colored graph is called point-determining if the underlying

graph is point-determining. A 2-colored graph is called semi-point-determining if all

vertices of the same color have distinct neighborhoods.

Note that the notion of co-point-determining 2-colored graphs is not interesting,

since any two adjacent vertices in a 2-colored graph are colored differently, so that

there is no vertex that could be adjacent to both of them.

We denote by P(X, Y ) the species of point-determining 2-colored graphs, by

Ps(X, Y ) the species of semi-point-determining 2-colored graphs, and by Pc(X, Y )

the species of connected point-determining 2-colored graphs.

Theorem 4.5.2. For species P(X, Y ), Ps(X, Y ) and Pc(X, Y ), we have

Ps(X, Y ) = (1 + X)(1 + Y ) E (Pc

≥2(X, Y )),(4.5.1)

P(X, Y ) = (1 + X + Y ) E (Pc≥2(X, Y )).(4.5.2)

Proof. Let G1 be a semi-point-determining 2-colored graph. We observe that

a connected component of G1 could be either a single vertex colored white, a single

vertex colored black, or a connected point-determining 2-colored graph with at least

two vertices. At the same time, G1 can have at most one isolated vertex colored

with each color, due to the fact that all vertices in G1 of the same color must have

distinct neighborhoods. Equation (4.5.1) follows by translating the above into species

equivalence.

Let G2 be a point-determining 2-colored graph. As in the above discussion we see

that a connected component of G2 could be either a single vertex colored white, a

single vertex colored black, or a connected point-determining 2-colored graph with at

117


Pc≥2

(X, Y )

EPs(X, Y )

1 + Y

1 + X

Pc≥2

(X, Y )Pc≥2

(X, Y )

Pc≥2

(X, Y )

Figure 4.20. Ps(X, Y ) = (1 + X)(1 + Y ) E (Pc≥2(X, Y )).

least two vertices. But this time, since the underlying graph of G2 is point-determining

graph, G2 can have at most one isolated vertex in all. Hence the term (1 +X)(1 +Y )

in (4.5.1) is replaced with the term 1 + X + Y in (4.5.2).

1 + X + Y

Pc≥2

(X, Y )

Pc≥2

(X, Y )

Pc≥2

(X, Y )

EP(X, Y )

Pc≥2

(X, Y )

Figure 4.21. P(X, Y ) = (1 + X + Y ) E (Pc≥2(X, Y )).

Theorem 4.5.3. For the species G (X, Y ) and Ps(X, Y ), we have

G (X, Y ) = Ps(E+(X), E+(Y )).

Proof. The proof use the same idea as the proof of Theorem 4.1.3. To be more

precise, given any 2-colored graph, we define equivalence relations on the vertex set

by setting two same-colored vertices to be equivalent if they have the same neigh-

borhoods, and get a new 2-colored graph whose vertex set is the set of equivalence

118


classes and the adjacency in the original graph is accordingly preserved. We observe

that the resulting new graph is a semi-point-determining 2-colored graph, and the

rest is straightforward.

We make use of the virtual species Γ defined in Section 2.4 again, and get from

Theorem 4.5.3 that

Ps(X, Y ) = G (Γ(X), Γ(Y )),

which, together with Equations 4.5.1 and 4.5.2, allows us to compute the associ-

ated series of the species of semi-point-determining 2-colored graphs, the species of

point-determining 2-colored graphs, and the species of connected point-determining

2-colored graphs as follows:

Ps(x, y) = 1 +

x

1!+

y

1!+ 2

x

1!

y

1!+ 2

x

1!

y2

2!+ 2

x2

2!

y

1!+ 10

x2

2!

y2

2!+ 24

x2

2!

y3

3!

+ 24x3

3!

y2

2!+ · · · ,

Ps(x, y) = 1 + x + y + 2xy + x2y + xy2 + 3x2y2 + 3x3y2 + 3x2y3 + · · · ,

ZPs(X,Y ) = 1 + (p1[x] + p1[y]) + (2p1[x]p1[y]) + (p21[x]p1[y] + p1[x]p2

1[y])

+

(1

2p2[x]p2[y] +

5

2p2

1[x]p21[y]

)

+ (p1p2[x]p2[y] + p2[x]p1p2[y] + 2p31[x]p2

1[y] + 2p21[x]p3

1[y]) + · · · .

P(x, y) = 1 +x

1!+

y

1!+

x

1!

y

1!+ 2

x

1!

y2

2!+ 2

x2

2!

y

1!+ 6

x2

2!

y2

2!+ 24

x2

2!

y3

3!

+ 24x3

3!

y2

2!+ · · · ,

P(x, y) = 1 + x + y + xy + x2y + xy2 + 2x2y2 + 3x3y2 + 3x2y3 + · · · ,

ZP(X,Y ) = 1 + (p1[x] + p1[y]) + (p1[x]p1[y]) + (p21[x]p1[y] + p1[x]p2

1[y])

+

(1

2p2[x]p2[y] +

3

2p2

1[x]p21[y]

)

119


+ (p1p2[x]p2[y] + p2[x]p1p2[y] + 2p31[x]p2

1[y] + 2p21[x]p3

1[y]) + · · · .

Pc(x, y) =

x

1!+

y

1!+

x

1!

y

1!+ 4

x2

2!

y2

2!+ 6

x2

2!

y3

3!+ 6

x3

3!

y2

2!+ · · · ,

Pc(x, y) = x + y + xy + x2y2 + x3y2 + x2y3 + · · · ,

ZPc(X,Y ) = (p1[x] + p1[y]) + (p1[x]p1[y]) + (p21[x]p2

1[y])

+

(1

2p1p2[x]p2[y] +

1

2p2[x]p1p2[y] +

1

2p3

1[x]p21[y] +

1

2p2

1[x]p31[y]

)+ · · · .

Figure 4.22. Unlabeled point-determining 2-colored graphs with nomore than 5 vertices.

We can write down the molecular decomposition of a 2-sort species. For example,

Figure 4.22 gives the first terms of the molecular decomposition of P(X, Y ).

P(X, Y ) = 1 + (X + Y ) + XY + (X + Y )(XY ) + (E2(X)E2(Y ) + X2Y 2)

+[2(X + Y )E2(XY ) + X3Y 2 + X2Y 3

]+ · · · ,

where the terms 2(X +Y )E2(XY )+X3Y 2 +X2Y 3 correspond to the unlabeled point-

determining 2-colored graphs with 5 vertices as shown in Figure 4.23.

120


XE2(XY )Y E2(XY )

X2Y 3

XE2(XY )

X3Y 2 Y E2(XY )

Figure 4.23. Unlabeled point-determining 2-colored graphs with 5 vertices.

121

APPENDIX A

Index of Species

0 empty species.

1 singleton set species.

X species of singletons.

Xn species of linear orders of order n.

Xn/A molecular species of (left) cosets of A in Sn.

A (X, Y ) 2-sort species of rooted trees.

C (X) species of oriented cycles.

Dn(X) molecular species Xn/Dn.

E (X) species of sets.

Ek(X) species of k-element sets.

Ek〈F 〉 exponential composition of species F of order k, for k ≥ 1.

E 〈F 〉 exponential composition of species F.

K (X) species of complete graphs.

G (X) species of (simple) graphs.

G c(X) species of connected graphs.

G (X, Y ) 2-sort species of 2-colored graphs, or bi-colored graphs.

G c(X, Y ) 2-sort species of connected 2-colored graphs.

122

CHAPTER A. INDEX OF SPECIES

Γ(X) virtual species of “connected (1 + X)-structures”, or

compositional inverse of E+.

L (X) species of linear orders.

N (X) species of (2-)rectangles.

N (k)(X) species of k-rectangles.

NN species of k-dimensional cubes on [N ], or E ⊡kn [N ], where N = nk.

OG(X) species associated to a graph G.

P(X) a) species of point-determining graphs (in chapter 4);

b) species of prime graphs.

Pc(X) species of connected point-determining graphs.

P(X, Y ) 2-sort species of point-determining 2-colored graphs.

Ps(X, Y ) 2-sort species of semi-point-determining 2-colored graphs.

Pc(X, Y ) 2-sort species of connected point-determining 2-colored graphs.

Φ(X) species of permutations.

Ψ(k)(X) species of partitions into k blocks .

Q(X) species of co-point-determining graphs.

Qc(X) species of connected co-point-determining graphs.

R(X) species of bi-point-determining graphs.

Rc(X) species of connected bi-point-determining graphs.

S (X) species of phylogenetic trees.

S ∗(X) species of alternating phylogenetic trees, or yoke-chains, or

series-parallel networks.

123

APPENDIX B

General Notations

A > B product of groups A and B acting on the set [m + n].

A × B product of groups A and B acting on the set [mn].

B ≀ A wreath product of groups A and B.

BA exponentiation group of A and B.

aut(G) automorphism group of a graph G.

B category of finite sets with bijections.

Bk category of k-sets with bijective multifunctions.

bn number of unlabeled prime graphs of order n.

C set of unlabeled connected graphs.

CM monoid algebra associated with a free commutative monoid M.

ck(λ) number of parts of length k in a partition λ.

cn number of labeled connected graphs of order n.

cn number of unlabeled connected graphs of order n.

c. t.(σ) cycle type of a permutation σ.

dλ number of permutations with cycle type λ.

Dn dihedral group of order n.

D(F ) Dirichlet exponential generating function of a species F.

D(G) Dirichlet exponential generating function of a graph G.

124

CHAPTER B. GENERAL NOTATIONS

E(G) edge set of a graph G.

gcd(k, l) greatest common divisor of integers k and l.

f1 ∗ f2 image of f2 under the operator obtained by substituting the

operator Ir for the variables pr in f1.

F (x) exponential generating series of a species F.

F (x) type generating series of a species F.

F1 + F2 sum of species F1 and F2.

F1 · F2 product of species F1 and F2.

F1 F2 composition of species F1 and F2.

F1 ⊡ F2 arithmetic product of species F1 and F2.

G1 ⊙ G2 Cartesian product of graphs G1 and G2.

λi ith part of the partition λ = (λ1, λ2, . . . ), arranged in

weakly decreasing order.

Kk,l complete bipartite graph on the set [k, l].

l(G) number of vertices in a graph G.

L(G) number of graphs isomorphic to G with vertex set V (G).

lcm(k, l) least common multiple of integers k and l.

N set of natural numbers.

Ω Q-algebra generated by the operators Ik.

P set of prime numbers.

P set of unlabeled prime graphs with respect to the Cartesian

multiplication.

125

CHAPTER B. GENERAL NOTATIONS

pn power sum symmetric function of order n.

pλ power sum symmetric function indexed by the partition λ.

pλ ⊠ pµ an operation defined by Definition 1.2.3.

Parn set of partitions of n.

Parkn set of sequences of k partitions of n.

Q set of rational numbers.

R ring of polynomials in the variables p1, p2, . . . with the operation ⊡ .

Sn symmetric group of order n.

V (G) vertex set of a graph G.

zλ number of permutations commute with a permutation of cycle type λ.

Z(A) Polya’s cycle index polynomial of a permutation group A.

ZF cycle index of F.

Z(G) = ZOGcycle index of the species associated to the graph G.

126

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