Counting Connected Digraphs with...

Counting Connected Digraphs with Gradings

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

Jonah Ostroff

August, 2013

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 Jonah Ostroff’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

Malcolm Watson, Dean of Arts and Sciences

Dissertation Committee:

Ira Gessel, Dept. of Mathematics, Chair.

Olivier Bernardi, Dept. of Mathematics

Vidya Venkateswaran, Dept. of Mathematics, MIT

c© Copyright by

Jonah Ostroff

2013

Acknowledgments

I would like to thank Ira for years of patience, guidance, and insight. I am also

grateful to Olivier and Vidya for being on my committee, to Janet and Tony for their

administrative support, to my classmates for the camaraderie and commiseration, and

to Susan, whose mentoring is largely responsible for the preservation of my sanity.

iv

Abstract

Counting Connected Digraphs with Gradings

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

University, Waltham, Massachusetts

by Jonah Ostroff

We define two structures on directed graphs called gradings and fencings, and find

a weighted correspondence between digraphs endowed with these structures and di-

graphs with certain connectivity conditions. We then show how this correspondence

may be used to enumerate various classes of strongly connected, initially connected,

and initially-finally connected graphs via straightforward combinatorial arguments,

and how it relates to some previously observed exponential generating function iden-

tities.

v

Contents

Introduction 1

Chapter 1. Digraphs, Graded and Fenced 4

1.1. Definitions 4

1.2. The Exponential Formula 8

1.3. Labeled Digraphs 9

1.4. Graded and Fenced Digraphs 11

1.5. The Involution Φ 13

1.6. The Involution Ψ 14

1.7. Level-Decomposability 18

1.8. Evaluating ΞD(n) 19

1.9. Computation of LD(a, b) 21

1.10. A Graphic Generating Function Approach 25

1.11. Unlabeled Digraphs 26

1.12. Another Approach for Strong Automata 27

Chapter 2. Other Connectivity Results 32

2.1. Restricted Fencings 32

2.2. Initially Connected Digraphs 33

2.3. W -initially connected Digraphs 35

2.4. Initially-Finally Connected Digraphs 37

2.5. Finally Connected Digraphs 41

vi

2.6. Computations 43

2.7. Digraphs with an Even Number of Edges 43

2.8. A Graded Interpretation of Some Generating Function Identities 45

Chapter A. Level Decomposability 49

A.1. Level Charts 49

A.2. Characterization of Level-Decomposable Sets 52

Chapter B. Tables 55

Bibliography 61

vii

Introduction

A directed graph is strongly connected if there is a path from every vertex to every

other vertex. In [5], Liskovets used combinatorial methods to derive a set of three

recurrences to count sn, the number of strongly connected digraphs with vertex set

{1, 2, 3, . . . , n} having no loops or multiple edges. In [12], Wright used the results of

Liskovets to derive a simpler set of recurrences:

ηn = −n∑i=1

(n− 1

i− 1

)siηn−i

ηn = −n∑i=1

(n

i

)2i(n−i)ηn−i.

Wright did not have a combinatorial interpretation for the numbers ηn, though a more

elegant explanation of the recurrences was later provided in [7].

We’re interested in enumerating many classes of strongly connected digraphs, as

well as digraphs with other connectivity conditions. We’ll find a generalization of

the above recurrences for different classes of digraphs. In doing so, we’ll see that the

numbers ηn (and their generalizations) actually count two different types of objects,

and that the two recurrences correspond to the two interpretations of these numbers.

As a motivating example, we’ll look at the case of oriented simple graphs, digraphs

with no loops in which there is at most one edge between any pair of vertices u and

v. We’ll count two types of these graphs: totally cyclic digraphs and graded digraphs.

1

CHAPTER 0. INTRODUCTION

First, a totally cyclic digraph is simply a disjoint union of strongly connected

digraphs. Consider the case of totally cyclic oriented simple graphs with vertex set

{1, 2, 3}. There are only three such graphs, as shown in Figure 0.1.

1

2 3

1

2 3

1

2 3

Figure 0.1. Three totally cyclic orientations of simple graphs on three vertices.

Next, a graded digraph is a directed graph in which the vertices are partitioned

into levels such that no edge leads from a higher level to a lower one. Again restricting

our focus to the set of oriented simple graphs on three vertices, there are 147 possible

graded digraphs. Figure 0.2 classifies these digraphs by the sizes of the levels:

6× 23

possibilities3× 22 × 3

possibilities3× 3× 22

possibilities1× 33

possibilities

Figure 0.2. All possible graded orientations of simple graphs on three vertices.

For example, when the vertices are all at different levels, there are 3! = 6 ways to

decide which vertex is at which level, and 23 ways to decide whether or not to draw

each of the possible upward-pointing edges. In the second case, there are 3 ways to

decide which vertex is at the lowest level, 22 ways to decide whether or not to draw

each of the upward edges, and 3 possibilities for the possible edge between the two

vertices at the higher level.

2

CHAPTER 0. INTRODUCTION

Obviously, then, the totally cyclic digraphs and graded digraphs are not in bijec-

tion with each other. However, we can assign positive and negative weights to each of

these objects to find a helpful correspondence. For totally cyclic digraphs, we assign

a positive weight when the number of strongly connected components is even, and a

negative weight when that number is odd. For graded digraphs, we assign a positive

weight when the number of levels is even, and a negative weight when it is odd.

In our example, the total weight of the totally cyclic digraphs is −3, since each

graph in Figure 0.1 has an odd number of components. For the graded digraphs, the

sum of the weights is −(6× 23) + (3× 22 × 3) + (3× 3× 22)− (1× 33) = −3.

This is no coincidence. For any set of digraphs D, this weighted equality holds.

We’ll formalize these definitions in Chapter 1, and then show how we can use them

to derive the above recurrences. In Chapter 2, we’ll use graded digraphs to count

initially, finally, and initially-finally connected digraphs, and see how other results

on the generating functions for such graphs can be easily derived through this corre-

spondence.

3

CHAPTER 1

Digraphs, Graded and Fenced

1.1. Definitions

A directed graph (or a “digraph”) is an ordered pair (V,A), where V is a set

of vertices and A is a set of ordered pairs of vertices called edges. For a digraph

D = (V,A), in an abuse of notation we will use V (D) to refer to its set of vertices V ,

and A(D) to refer to its set of edges A. For a particular edge e = (u, v) ∈ A(D), we

will sometimes call v the head of e and u the tail of e.

For nonnegative integers n, let [n] denote the set {1, 2, . . . , n}. (Note that [0] =

∅.) We are primarily concerned with labeled digraphs, in which the vertices of the

graph are labeled by elements of some set, usually [n]. The symmetric group Sn

acts naturally on such digraphs: for a permutation σ ∈ Sn, relabel the vertices by

replacing i with σ(i). If instead we wanted to count unlabeled digraphs, we would

identify all labeled digraphs in an orbit under this action.

In several instances, the fact that D is labeled will allow us to arbitrarily choose

a set of vertices by its label set. In particular, we will fix an ordering on the set of all

nonempty subsets of [n], so that when we must choose between several such subsets

in a collection, we can call one of them the “primary” subset from that collection.

For two distinct subsets S1 and S2 of [n] with minimum elements s1 and s2, we

say S1 ≺ S2 if any of the following hold:

• |S1| < |S2|.

• |S1| = |S2| and s1 < s2.

4

CHAPTER 1. DIGRAPHS, GRADED AND FENCED

• |S1| = |S2| and s1 = s2 and S1\{s1} ≺ S2\{s2}.

For a collection C of subsets of [n], we say that S ∈ C is the primary element of

C if it is the maximal element of C under the ordering ≺.

A subgraph of a digraph (V,A) is any digraph (V ′, A′) such that V ′ ⊆ V,A′ ⊆ A.

An induced subgraph of (V,A) is a digraph (V ′, AV ′) where AV ′ is the set of all edges

in A whose head and tail are both in V ′. The disjoint union of two digraphs (V1, A1)

and (V2, A2) is simply (V1 t V2, A1 t A2).

A loop in a digraph is an edge of the form (v, v) for some v. Sometimes we will

consider digraphs with multiple edges, in which A(D) is a multiset and may contain

a particular ordered pair any number of times.

The indegree of a vertex v is #{(u, v) ∈ A(D) : u ∈ V (D)}, the number of edges

whose head is v. Likewise, the outdegree of a vertex u is #{(v, u ∈ A(D) : u ∈ V (D)},

the number of edges whose tail is v.

Note that the existence of an edge (u, v) does not usually preclude the existence

of its opposing edge (v, u). When both of these edges are elements of A(D), we

sometimes refer to them jointly as a bidirectional edge, even though they are still

counted as separate edges for purposes of indegree and outdegree. When only one of

these is present, it is a one-directional edge. When neither is present, then u and v

have a missing edge.

At times we will consider graphs where the edges are labeled as well. Specifically,

for a vertex v with outdegree k, the k edges originating at v are labeled 1, . . . , k.

When multiple edges are not allowed, the number of edge labelings of a given digraph

is determined solely by the outdegrees of the vertices. With multiple edges, the

enumeration is more interesting. For example, in Figure 1.1, the edges of the first

5


graph can be labeled in two different ways, but those in the second can be labeled in

only one way.

1

2 3

1

2 3

Figure 1.1. Two graphs with the same outdegrees, but with differentnumbers of possible edge-labelings.

Sometimes we’ll also allow A(D) to contain free edges of the form (v,−). Free

edges contribute to the outdegree of v but do not contribute to the indegree of any

vertex.

For vertices u, v ∈ V (D), a directed path from u to v is a sequence of vertices

u = v1, v2, . . . , vk = v ∈ V (D) such that (vi, vi+1) ∈ A(D) for 1 ≤ i ≤ k − 1. We say

that v is reachable from u, denoted u v, if there exists a directed path from u to

v. A digraph D is strongly connected (or is a “strong digraph”) if for all u, v ∈ V (D),

u v.

Note that is reflexive, because the path with one vertex is still a path. It is

also transitive, because whenever u1 u2 and u2 u3, we can concatenate the

corresponding paths to find a path from u1 to u3. So defines a preorder on the

vertices of D.

We can also define an equivalence relation !, where u! v if both u v and

v u. The equivalence classes of vertices in a digraph under ! are called strongly

connected components. So a digraph is strongly connected if it consists of exactly one

strongly connected component.

A digraph is initially connected from vertex u if for all v ∈ V (D), u v. If the

vertex u is not specified and the digraph in question is labeled by [n], then “initially

connected” should be taken to mean “initially connected from 1”.

6


An undirected path from u to v is a path which need not necessarily follow the

directions of the edges in the graph. Specifically, it is a sequence of vertices u =

v1, . . . , vk = v such that for 1 ≤ i ≤ k − 1, at least one of (vi, vi+1) or (vi+1, vi) is in

A(D). A subset of a digraph W ⊆ V (D) is weakly connected if there exist undirected

paths between each pair of vertices in W .

A digraph is totally cyclic if is symmetric. If this is true, then there can never

be any edges between two strong components. In other words, D is totally cyclic if

and only if it can be written as the disjoint union of one or more strong digraphs.

The name totally cyclic comes from the equivalent characterization that every edge

is part of a cycle.

1

2

3

4

56

7

8

9

10

11

12

13

14

Figure 1.2. A totally cyclic graph on 14 vertices.

Our goal will be to enumerate, for a particular set of digraphs D, the number of

digraphs in D labeled by [n] satisfying some of the above properties. Oftentimes D

will be endowed with an R-valued weight function w : D → R for some ring R. In

that case, our goal will be to compute∑

D∈P w(D) for some subset of graphs P ⊆ D

satisfying a certain property.

7


The decomposability of the weight of a digraph into its components will be a useful

feature for much of this enumeration, but unfortunately it is one we do not always

have. Specifically, we say that D is strongly decomposable if the following conditions

hold:

• A totally cyclic digraph D is in D if and only if each of its strong components

are in D.

• Any relabeling of a graph in D with the same set of labels is also in D. When

D is weighted, these two graphs must have the same weights.

When these conditions are met, it will be easy to count strongly connected di-

graphs once we’ve enumerated totally cyclic digraphs, by using the exponential for-

mula, which will be explored in the next section.

Strongly connected labeled digraphs with no loops or multiple edges were first

counted by Liskovets in [5]. A simpler pair of recurrences were derived algebraically

from Liskovets’s results by Wright in [12], and a shorter proof was given by Robinson

in [7]. Robinson later answered the same question for automata (d-regular digraphs

with edge labelings) in [8]. In [11], Wright enumerated strongly connected tourna-

ments. In this chapter, we’ll develop our main theorem which allows us to generalize

all of these results, and also gives a simple combinatorial interpretation to the recur-

rences in [12].

1.2. The Exponential Formula

Let F be a family of objects with labels, and let Fn denote the members of F

whose label set is [n]. We’re particularly interested in the case where Fn is closed

under relabeling. That is, every permutation π ∈ Sn induces a permutation on Fn by

permuting the labels of each object along π.

8


If |Fn| is finite for each n, then we say that the exponential generating function

for F is the formal power series

∞∑n=0

|Fn|xn

n!.

In the case where F is weighted, for any S ⊆ F , we define |S| as the sum of the

weights of the elements of S.

For a family of objects F closed under relabeling, we define E(F) to be a new

family where an object in E(F)n is given by:

• A partition of [n] into nonempty sets A1, A2, . . . , Ak.

• For 1 ≤ i ≤ k, an object in F labeled by Ai.

When F is weighted, then the weight of an object in E(F) is taken to be the

product of the weights of its component objects from F .

The exponential formula says that if F (x) is the exponential generating function

for E(F), and g(x) is the exponential generating function for F , then

F (x) = exp g(x).

For more on the exponential formula, see [9] and [10]. The combinatorics of

labeled and unlabeled objects, and particularly the construction of E(F), is elegantly

formalized through categorical terms in [4] via the theory of combinatorial species.

1.3. Labeled Digraphs

For many sets of digraphs D, our main goal is to count the number of strongly

connected digraphs in D with vertex set [n]. We will denote the set of strongly

connected digraphs on D by Str(D), and the set of totally cyclic digraphs on D by

9


TC(D). For any set S of digraphs, we will let Sn denote the subset of S consisting of

digraphs whose vertex set is [n].

While it would be nice to find the size of TC(Dn) directly, it turns out to be more

tractable to consider a slightly modified sum instead. For a totally cyclic digraph D,

let c(D) denote the number of weak components of D. Our main result deals with

the sequence

ΞD(n) =∑

D∈TC(Dn)

(−1)c(D).

For a strongly decomposable set of digraphs D, it is easy to use ΞD(n) to derive

the number of strongly connected digraphs in Dn. Specifically, let CSD(n) denote the

number of digraphs in Str(Dn).

Theorem 1.3.1. Suppose D is strongly decomposable. Then

ΞD(n) = −n∑i=1

(n− 1

i− 1

)CSD(i)ΞD(n− i)

Proof. In a totally cyclic digraph on [n], the weak component containing the

vertex 1 is strongly connected. By summing over the possible sizes i of this component,

we find that there are(n−1i−1

)ways to pick which other vertices lie in that component

and CSD(i) ways to create a strong component on those vertices.

The remaining n− i also vertices comprise a totally cyclic digraph, and ΞD(n− i)

counts these possibilities. But because ΞD(n − i) does not account for the strong

component containing 1, we must use the factor −ΞD(n − i) instead to have the

correct sign. �

Note that this recurrence is closely related to the exponential formula. Suppose

we have

10


b(x) =∞∑n=0

ΞD(n)xn

n!

s(x) =∞∑n=0

CSD(n)

xn

n!

Then we can rewrite Theorem 1.3.1 as

ΞD(n+ 1) = −n∑i=0

(n

i

)CSD(i+ 1)ΞD(n− i)

which is equivalent to saying that

b′(x) = −s′(x)b(x).

Thus, b′(x)/b(x) = −s′(x), so s(x) = − log b(x).

Equivalently, b(x) = exp(−s(x)). In terms of the exponential formula, this makes

perfect sense: each totally cyclic digraphs is really the disjoint union of some set of

strongly connected digraphs whose label sets give a partition of [n], and its weight is

−1#components.

In the case where D is the set of all digraphs with no loops or multiple edges,

Theorem 1.3.1 is equivalent to the first of a pair of recurrences found by Wright in

[12]. For a combinatorial interpretation of the second of these recurrences, we turn

to another understanding of the numbers ΞD(n) introduced in the next few sections.

1.4. Graded and Fenced Digraphs

In this section, we introduce two new structures on digraphs. A graded digraph

with k levels is a digraph together with a grading: an assignment of a positive integer

height from 1 to k to each vertex of the graph, satisfying the following criteria:

11


• For 1 ≤ i ≤ k, the set of vertices with height k is nonempty.

• No edge leads from a vertex of height i to a vertex of height less than i.

If D is a graded digraph, we let `(D) denote the number k of nonempty levels.

Note that several gradings are possible for a particular digraph unless that digraph

is strongly connected.

Second, a fenced digraph is a digraph together with a fencing (FM , Fm). The

sets FM and Fm consist of nonempty, possibly intersecting, subsets of the vertices

respectively called major and minor fences, satisfying the following criteria:

• No edge leads from a vertex within a fence to a vertex outside that fence.

• Every vertex is in exactly one major fence.

• Every minor fence is a proper subset of a major fence.

• As sets, any two fences are distinct.

If D is a fenced digraph, we let f(D) denote the total number of major and minor

fences in D. As with gradings, a digraph which is not strongly connected has several

possible fencings. An example of a graded digraph and a fenced digraph can be seen

in Figure 1.3.

1

2

3

4

5

6 7

1

2

3

4

5

67

height 1

height 2

height 3

height 4

Figure 1.3. A graded digraph with four levels, and a fenced digraphwith six fences

12


If D is a set of digraphs, we use TC(D) to denote the set of totally cyclic digraphs

in D, Gr(D) to denote the set of gradings of digraphs in D, and Fen(D) to denote

the set of fencings of digraphs in D.

These three types of digraphs are related in a particularly helpful way.

Theorem 1.4.1. Let D be an arbitrary set of directed graphs. Then for any n,

∑D∈TC(Dn)

(−1)c(D) =∑

D∈Fen(Dn)

(−1)f(D) =∑

D∈Gr(Dn)

(−1)`(D).

We prove this by canceling terms from the middle sum in two different ways, by

defining two sign-reversing involutions Φ and Ψ on subsets of Fen(D).

1.5. The Involution Φ

In a particular fenced digraph, say that a set of vertices is an admissible fence if

it exists as a minor fence, or if its inclusion as a minor fence would still yield a valid

fencing. Notice that the set of admissible fences in a fencing is determined solely by

the underlying digraph and the set of major fences.

On the set of major fences with admissible minor fences, the involution Φ is defined

as follows:

For a fenced digraph D with fencing (FM , Fm) , consider the set of all admissible

fences on D. Take the primary admissible fence, f1, and either add or delete it

from the set of minor fences. That is to say, Φ(D) is the same digraph with the

same fencing, but with Fm replaced by Fm 4 {f1}, where 4 denotes the symmetric

difference. Note that this is indeed an involution, since the set of admissible fences is

not altered by the addition or deletion of minor fences, and so a second application

of Φ will delete f1 (if it was added the first time) or add it back in (if it was deleted

13


the first time). Furthermore, it is sign-reversing, because exactly one fence is created

or removed.

After the cancellation of pairs of fencings matched by Φ what remains is the set

of fenced digraphs with no admissible minor fences. This means that every major

fence—which necessarily creates a partition of the graph—is strongly connected: if it

were not strongly connected, then a minor fence could exist inside it. These strongly

connected components are also the weakly connected components, since no edge may

leave a major fence. So what remains from the cancellation through Φ is exactly the

set of totally cyclic digraphs, signed according to the number of components.

1.6. The Involution Ψ

This involution pairs off all fenced digraphs except for a particular type: fencings

such that for any two fences f1 and f2, either f1 ⊂ f2 or f2 ⊂ f1. There is an easy

correspondence between such fencings and gradings: the height of a vertex i is simply

the number of fences containing i.

1 2

3

4

5

6

7

8

1 2

3

4

5

6

7

8

height 4

height 3

height 2

height 1

Figure 1.4. A graded digraph, and the corresponding nested fencing

14


We will construct a sign-reversing involution Ψ on all fencings not having this

property. Let the triple (D,FM , Fm) denote a digraph D together with a valid set of

major fences FM and minor fences Fm. We will pair it with a similar triple (D,F ′M , F′m)

having one more or one fewer fence.

First, if the number of major fences is greater than one, we can pair this triple

with the same digraph D with F ′M = {[n]} and F ′2 = FM ∪ Fm, i.e. the major fences

merged into one and replaced with minor fences. Conversely, if there is exactly one

major fence and a set Fmax ⊆ Fm of minor fences which partitions the vertices of the

graph, with the property that for all fm ∈ Fm there exists an f ′m ∈ Fmax such that

fm ⊆ f ′m, we can perform the reverse operation: Ψ(D,FM , Fm) = (D,Fmax, Fm\Fmax).

The involution involves exchanging k major fences for k minor fences and one major

fence, so the paired triples have opposite sign. The correspondence is shown in the

first row of Figure 1.5.

Next, suppose there is exactly one major fence and no set of minor fences which

partitions the graph as in the previous case. We now look for a pair of minor fences

with non-empty intersection, neither of which is a subset of the other. Of all such

pairs of fences, choose the pair f1 and f2 such that for all other pairs f ′1, f′2, either

f ′1 ≺ f1 or f1 = f ′1 and f ′2 ≺ f2. Note that since f1 and f2 are both fences, every

edge originating in f1 ∩ f2 must also lead to f1 ∩ f2, so we can add or subtract f1 ∩ f2

from F2 to change the sign of the triple and still have a valid set of fences for D. So,

specifically, we have F ′1 = F1 = {[n]}, F ′2 = F2 4 {f1 ∩ f2}. This is an involution

because, in the case where a new minor fence f3 is created, it is a subset of f1 and f2

and so f3 ≺ f1 and f3 ≺ f2, which means that f1 and f2 will still be selected when Ψ

is repeated. A single fence is added or removed, so again the sign is changed. This

pairing is illustrated in the second row of Figure 1.5.

15


Figure 1.5. The involution Ψ, as it acts on three pairs of fencings.

Finally, consider the case where there is one major fence and no maximal set of

minor fences which partitions the graph and no pair of intersecting non-nesting fences.

We will look for a set of disjoint fences. Specifically, we will choose the “outermost”

set of disjoint fences: a set of two or more disjoint fences so that any other fence is

either contained in one of them, or contains all of them. Because each of these is a

fence, we can add or subtract their union fU as a minor fence. Thus, F ′1 = F1 = {[n]},

and F ′2 = F2 4 {fU}. Once again a single fence is added or deleted. An example is

shown in the third row of Figure 1.5.

What Ψ does not act on are the triples comprising a single major fence and a set

of minor fences such that any pair of them is nested, or in other words a chain of

fences. We assign a vertex contained in i of those fences (including the major fence)

16


to height i. Then, indeed, we’re left with the structure we want: a digraph in which

no edge leads from a vertex of height i to a vertex of height less than i, because that

would necessitate leaving a fence. We are now ready to prove Theorem 1.4.1.

Proof of Theorem 1.4.1. We have constructed two involutions on the terms

of the central sum. The involution Φ cancels out all terms except those which cor-

respond to totally cyclic digraphs, yielding the left-hand side. The involution Ψ

cancels out all terms except those which correspond to graded digraphs, yielding the

right-hand side. �

In fact, because the images of Φ and Ψ are always different fencings of the same

underlying digraph, this equality still holds if D is endowed with a weight function

w.

Proposition 1.6.1. Suppose D is a set of digraphs with weights w. Then

∑D∈TC(Dn)

(−1)c(D)w(D) =∑

D∈Fen(Dn)

(−1)f(D)w(D) =∑

D∈Gr(Dn)

(−1)`(D)w(D).

We can also prove the first equality by collecting the fencings of D according to

the set of major fences. First, we write the middle sum as

∑D∈D

∑FM

∑Fm

(−1)|FM |+|Fm|w(D),

where the second sum is taken over all valid choices of major fences for D, and the

third sum is taken over all valid choices of minor fences for D, each of which is

a proper subset of a major fence in FM . Note that for a given choice of FM , each

admissible minor fence may independently be chosen to be in Fm or not, which means

it contributes a factor of −1 (if it is chosen to be in Fm) or +1 (if it is not). Let

Adm(D,FM) denote the number of admissible minor fences for a digraph D with

17


major fences FM . Then we can rewrite the sum as

∑D∈D

∑FM

(−1)|FM |(1− 1)Adm(D,FM )w(D).

So each choice of D and FM has a nonzero contribution to the sum if and only

if Adm(D,FM) = 0, which is to say that there are no admissible minor fences. As

shown in Section 1.5, a major fence has no admissible minor fences if and only if it is

strongly connected, so this sum counts digraphs which can be partitioned into disjoint

strongly connected components, weighted positively or negatively according to by the

number of such components. This is exactly the sum on the left in Proposition 1.6.1.

1.7. Level-Decomposability

In many cases, we’ll see that graded digraphs are very simple to count. There is

a particular condition that makes this enumeration easy, illustrated in the following

example.

Suppose we want to count all graded digraphs on [5] with no loops or multiple

edges, with two vertices at the first level, two at the second, and one at the third.

Before we draw the edges or label the vertices, such a graph looks like the left side of

Figure 1.6.

2 3

1 5

4

2 3

1 5

4

Figure 1.6. Construction of a graded digraph on [5].

18


Next, we label the vertices. At the moment, the vertices at each level are indis-

tinguishable, so we can choose the labeling in 5!/(2! 2!) = 30 ways. Once the vertices

are labeled, we choose which edges to draw. In the middle of Figure 1.6, dotted

arrows show each possible edges that might be drawn. For these 12 possible edges,

our choices of whether or not to include each edge can be made independently in 212

ways. A possible resolution is shown on the right side of Figure 1.6.

Notice that once we know how many vertices are at each level, it is easy to

count how many ways the outgoing edges from each level can be drawn. When this is

possible for a particular family of digraphs D, we’ll use LD(a, b) to denote the number

of ways to draw the outgoing edges from a level with a vertices when there are b− a

vertices at higher levels. In this case, LD(a, b) = 2a(b−1). Whenever such a function

exists, we’ll say that D is level-decomposable. A more formal treatment of this idea

is given in Appendix A.

1.8. Evaluating ΞD(n)

When D is level-decomposable and we can compute L(a, b), the sum in Theorem

1.4.1 is particularly easy to evaluate. The most straightforward way is to count graded

digraphs by summing over the possible assignments of heights to the vertices. Let

c = (c1, c2, . . . , ck) be a composition of n: a sequence of positive integers such that∑i ci = n. For convenience, we’ll also define the associated sequence of partial sums

ti =∑i−1

j=1 cj. Then the number of graded digraphs with ci vertices at level i is given

by (n

c1, c2, . . . , ck

)∏i

L(ci, n− ti).

19


This means that the total signed number of graded digraphs on [n] is given by

Ξ(n) =∑c

(−1)k(

n

c1, c2, . . . , ck

)∏i

L(ci, n− ti).

Summing over compositions, unfortunately, can be rather unwieldy. An equivalent

definition is given by a determinant formula:

(1) Ξ(n) = (−1)nn!∣∣aij∣∣, where aij =

L(j − i+ 1, n− i+ 1)

(j − i+ 1)!, if j > i− 1

1, if j = i− 1

0, if j < i− 1

For example,

Ξ(3) = (−1)33!

∣∣∣∣∣∣∣∣∣∣∣∣∣

L(1, 3)

1!

L(2, 3)

2!

L(3, 3)

3!

1L(1, 2)

1!

L(2, 2)

2!

0 1L(1, 1)

1!

∣∣∣∣∣∣∣∣∣∣∣∣∣.

Expanding across the first row of the determinant gives a recurrence:

(2) Ξ(n) = −n∑i=1

(n

i

)L(i, n)Ξ(n− i) Ξ(0) = 1

This recurrence can also be shown directly:

Proof. We will count the total number of graded digraphs on n vertices (weighted

by (−1)#levels) by summing over the possible values of i, the number of vertices at

height 1. For each i, there are(ni

)ways to pick which vertices are at height 1, L(i, n)

20


ways to draw the outgoing edges from the vertices of height 1, and Ξ(n − i) ways

to draw the rest of the graph, weighted by (−1)#levels−1. The factor −1 corrects this

sign. �

In the case whereD is the set of digraphs with no loops or multiple edges, Equation

(2) is equivalent to the second recurrence from [12]. So Wright’s results correspond

to the twin interpretations of ΞD(n), first as totally cyclic digraphs and second as

gradings. This combinatorial interpretation of the recurrences is new.

1.9. Computation of LD(a, b)

The values of LD(a, b) are often easy to compute through simple combinatorial

arguments.

Proposition 1.9.1. If D is the set of digraphs with no loops or multiple edges,

where missing edges are weighted α, one-directional edges are weighted β, and bidi-

rectional edges are weighted γ, then FD(a, b) = (α + 2β + γ)a(a−1)/2(α + β)a(b−a).

Proof. For each pair of vertices at the same level, the sum of the possible weights

contributed by edges in that pair is α+ 2β+γ, since there is either no edge, one edge

in either direction, or a bidirectional edge. For each pair of vertices at different levels,

the possibilities are simply no edge or a one-directional edge. These choices are all

taken independently. �

Note that in order for D to be strongly decomposable, we need α = 1, since

otherwise the weight of a disjoint union is not the product of the weights of its

components. On the other hand, if α = 0 then there exists only one weak component,

so every totally cyclic graph is strongly connected, and therefore we don’t need the

exponential formula in the first place. For other nonzero values of α, we should rescale

the other weights so that α = 1.

21


Corollary 1.9.2. If D is the set of digraphs with no loops or multiple edges,

then LD(a, b) = 2a(b−1).

Proof. This is a special case of Proposition 1.9.1, where α = β = γ = 1. �

Corollary 1.9.3. If D is the set of digraphs with no loops or multiple edges and

with edges weighted by β, then LD(a, b) = (1 + β)a(b−1).

Proof. This is a special case of Proposition 1.9.1, where α = 1, γ = β2. �

Corollary 1.9.4. If D is the set of orientations of simple graphs, then LD(a, b) =

3a(a−1)/22a(b−a).

Proof. This is a special case of Proposition 1.9.1, where α = β = 1, γ = 0. �

Corollary 1.9.5. If D is the set of tournaments, then LD(a, b) = 2a(a−1)/2.

Proof. This is a special case of Proposition 1.9.1, where α = γ = 0, β = 1. �

Note that in the case of Corollary 1.9.5, −ΞD(n) counts strongly connected tour-

naments directly because α = −1. Such tournaments have been previously counted

in [11].

We can also make an analogue of Proposition 1.9.1 with multiple edges allowed,

but it’s not immediately obvious what the weights should be. For example, if there

are 5 edges from u to v and 7 edges from v to u, how do we weight these? One choice

is to weight this as β2γ5.

Proposition 1.9.6. Let D be the set of digraphs with missing edges weighted by

α, where a pair of vertices with i > 0 vertices in one direction and i + j ≥ i vertices

in the other direction is given weight βjγi. Then

LD(a, b) =

(α +

γ

1− γ+

1

1− γ2β

1− β

)(a2)(

α +β

1− β

)a(b−a)22


Proof. For each of the(a2

)pairs of vertices at the same level, we may either

have no edges between them (with weight α), or the same number of edges in each

direction (with weight γ + γ2 + · · · = γ/(1− γ)), or, in two ways, we may have zero

or more edges in each direction plus an extra number of edges in one direction, with

weight (1 + γ + γ2 + · · · ) · (β + β2 + · · · ).

For each pair of vertices at distinct heights, we either have no edges or one or more

one-directional edges, so the total weight is α+β+β2 +β3 + · · · = α+β/(1−β). �

Proposition 1.9.7. If D is the set of digraphs with edges weighted by β, then

LD(a, b) =

(1

1− β

)ab.

Proof. For each of the ab pairs of vertices u and v for which an edge from u to

v is allowed, the sum of the possible weights contributed by any edges from u to v is

given by 1 + β + β2 + · · · = 1/(1− β). Equivalently, this is a corollary of Proposition

1.9.6 with α = 1, γ = β2. �

Proposition 1.9.8. If D is the set of d-regular digraphs (with multiple edges

allowed), then LD(a, b) =

(b+ d− 1

d

)a.

Proof. Each of the a vertices at a particular level can have an outgoing edge

toward b possible vertices, and it must lead to exactly d of them, perhaps with repe-

tition. �

Proposition 1.9.9. If D is the set of d-regular digraphs with no multiple edges,

then LD(a, b) =

(b

d

)a.

Proof. Each of the a vertices at a particular level can have an outgoing edge

toward b possible vertices, and it lead to exactly d of them. �

23


The next few results deal with automata and free edges. A d-regular automaton is

a labeled digraph in which each vertex has d outgoing edges labeled 1, . . . , d. A free

edge in such a graph is an edge of the form (v,−), an edge with a tail but no head.

These free edges contribute to the outdegree of a vertex, but not to any indegree.

Proposition 1.9.10. If D is the set of d-regular automata (without free edges),

then LD(a, b) = bda.

Proof. Each of the a vertices at a particular level has d distinguishable outgoing

edges, each of which has b possible endpoints with repetition allowed. �

Proposition 1.9.11. If D is the set of d-regular automata with free edges weighted

by α, then LD(a, b) = (b+ α)da.

Proof. As in Proposition 1.9.10, we must choose one of b possible endpoints for

each of the da edges, except that an extra option with weight α is available. �

Proposition 1.9.12. If D is the set of d-regular automata with no multiple edges,

then LD(a, b) =

((b

d

)d!

)a.

Proof. This is analogous to 1.9.10, but the choices must be distinct. �

A nondeterministic automaton on d letters is a labeled digraph in which vertex

has at least one outgoing edge labeled i for each i ≤ d. Such a digraph may fea-

ture multiple edges as long as they have differing labels. Note that unlike ordinary

automata, such digraphs need not be d-regular.

Proposition 1.9.13. If D is the set of nondeterministic automata on d letters,

then LD(a, b) = (2b − 1)da.

24


Proof. For each of the a vertices at a given level and d possible edges labels, we

must choose a nonempty set of vertices at the same or higher level to which we can

draw edges with that label. There are 2b − 1 such sets. �

1.10. A Graphic Generating Function Approach

In the previous section, the choices of D in Propositions 1.9.1 and 1.9.6 (as well

as those in their corollaries) have the property that LD(a, b) = u(a2)va(b−a) for some

constants u and v. In these cases, we will be able to compute the coefficients of

bD(x) =∞∑n=0

ΞD(n)xn

n!.

But first, we need to turn to the language of graphic generating functions. Let v

be some nonzero constant. The graphic generating function for a sequence an is the

formal power series

ga(x) =∞∑n=0

anxn

v(n2)n!

.

For more on graphic generating functions, see [7] (in which they are known as

“special generating functions”) and [1]. A crucial property of these power series is

that if an and bn are sequences with graphic generating functions ga and gb, then the

product ga(x) · gb(x) has coefficients given by

cn =n∑i=0

(n

i

)vi(n−i)aibn−i .

This convolution might look familiar: if LD(a, b) = u(a2)va(b−a), then Equation (2)

can be rewritten in the following form:

25


(3)n∑i=0

(n

i

)vi(n−i)u(i

2)ΞD(n− i) =

1 if n = 0

0 if n ≥ 1

Notice that the left side is exactly the definition of cn above, where an = u(n2) and

bn = ΞD(n). So we have

[∞∑n=0

u(n2) xn

v(n2)n!

][∞∑n=0

ΞD(n)xn

v(n2)n!

]= 1.

There is an analogue of the exponential formula for graphic generating func-

tions, but unfortunately it uses a decomposition into initially connected components

(treated in the next chapter) rather than our strongly connected components. So to

compute the generating function for strongly connected digraphs, we must extract

the coefficients ΞD(n) and compute the logarithm of the corresponding exponential

generating function.

1.11. Unlabeled Digraphs

It should be noted that Theorem 1.4.1 does not extend to the case whereD consists

of unlabeled digraphs. For example, there is one totally cyclic orientation of a simple

graph on two unlabeled vertices: namely, the digraph with no edges. However, there

are four graded orientations of simple graphs on two vertices, as shown in Figure 1.7:

height 1

height 2

Figure 1.7. The four graded orientations of simple digraphs on twounlabeled vertices.

26


An unlabeled analogue of Theorem 1.4.1 would suggest that summing (−1)h(D)

over these four graded digraphs gives the same result as summing (−1)c(D) over the

totally cyclic digraphs, but that is not the case: the sole totally cyclic digraph con-

tributes 1 to that sum, while these four gradings give a total of (−1)+(−1)+1+1 = 0.

This is not surprising, since both Φ and Ψ relied on the labeling of D. In the next

chapter we will see a few cases in which our results can be used to count unlabeled

digraphs, but unfortunately these instances are rare.

1.12. Another Approach for Strong Automata

In this section, we will see another way of counting strong automata which is

closely related to our earlier recurrences. In particular, we will find a combinatorial

interpretation for another recurrence obtained from the determinant formula, in which

we expand down the last column rather than across the first row.

In a (not necessarily strong) digraph D, a source strong component is a subgraph

C ⊆ D such that C is a strong digraph and any edge pointing into C must originate

in C. (We do allow edges from C to the rest of the graph.)

Let A be the set of d-regular automata with free edges allowed and weighted by

α. Let s(x, α) be the exponential generating function for strong digraphs in A. We

are interested in the coefficients Ξ(n, α, t) in

ets(x,α) =∞∑n

Ξ(n, α, t)xn

n!.

Note that when t = −1, this is equivalent to the previous definition of ΞA(n). We

have

Ξ(n, α, t) =∑

D∈TC(An)

tc(D)α#free edges

27


Theorem 1.12.1. For an automaton D, let σ(D) denote the number of source

strong components of D and let ϕ(D) denote the number of free edges in D. Then we

haven∑i=0

(n

i

)Ξ(i, α + n− i, t)(α + n− i)d(n−i) =

∑D∈An

(1 + t)σ(D)αϕ(D).

Proof. We count in two ways the set of ordered pairs (D,S), where D ∈ An and

S is some set of source strong components of D, weighted by t|S|αϕ(D).

Let V (S) denote the set of vertices in all source strong component in S. We’ll

count the pairs (D,S) in which |V (S)| = i. We choose V (S) in(ni

)ways. Next, we

must attach edges from those i vertices to each other and to the other n− i vertices.

We first examine D|V (S), that is to say, the digraph D restricted to the vertices of

S. In this restriction, edges which originate in V (S) but end outside V (S) are treated

as free edges in D|V (S).

So a free edge in D|V (S) might have been a free edge in D (in which case it should

have weight α), or it may have been connected to one of the vertices in V (D)\V (S)

(in which case there are n− i ways to attach it). Because of this, a free edge in D|V (S)

should be counted with weight α+n− i, and so the sum over all ways to attach edges

to the vertices in V (S) has weight Ξ(i, α+ n− i, t). Finally, the d(n− i) edges from

the vertices of V (D)\V (S) can either be free or point to a vertex in V (D)\V (S), so

those edges can be assigned in (α + n − i)d(n−i) ways. Putting this all together, we

see that the sum over the pairs (D,S) is

n∑i=0

(n

i

)Ξ(i, α + n− i, t)(α + n− i)d(n−i).

Second, we can count these pairs by summing over all automata D on [n] and ask-

ing how many times each one is counted. In this case, each source strong component

of D either may or may not be chosen as a member of S, and so contributes either a 1

28


or a t to the weight. In either case, the number of free edges is always ϕ(D). So each

digraph D ∈ An contributes (1 + t)σ(D)αϕ(D) to the sum, completing the proof. �

We’re particularly interested in the case where t = −1, so 1 + t = 0. Usually we’ll

suppress the t in this case, so we write Ξ(n, α) to mean Ξ(n, α,−1). For n > 0 we’re

guaranteed to have at least one source strong component, and so we get

(4)n∑i=0

(n

i

)Ξ(i, α + n− i)(α + n− i)d(n−i) = 0, n > 0.

Solving for Ξ(n, α) yields the recurrence

(5) Ξ(n, α) = −n−1∑i=0

(n

i

)Ξ(n− i, α + i)(α + i)id, Ξ(0, α) = 1.

For any level-decomposable set of digraphs D, we can replace (α + b)da with

LD(a, b + α), and define ΞD(n, α) as in Equation (5). This recurrence is closely

related to the determinant formula for Ξ(n) that we found earlier. In fact, it simply

corresponds to another expansion: Previously we found Equation (2) by expanding

across the first row, because the determinants of the resulting minors corresponded

to the values of Ξ(k) for k < n.

We could expand down the last column instead, but now the minors do not quite

look like the original matrix. Here’s the minor obtained by taking the ith entry from

the bottom in the last column, as well as the necessary 1s in the i− 1 rows below it:

29


L(1, n)

1!

L(2, n)

2!· · · L(n− i, n)

(n− i)!

1L(1, n− 1)

1!· · · L(n− i− 1, n− 1)

(n− i− 1)!

......

. . ....

0 0 · · · L(1, i+ 1)

1!

This is the matrix from the expression for Ξ(n− i), except that each occurrence of

L(a, b) is replaced by L(a, b+ i). We can include this substitution as a new parameter

in the definition of Ξ(n):

(6)

ΞD(n, α) = (−1)nn!∣∣aij∣∣, with aij =

L(j − i+ 1, n− i+ 1 + α)

(j − i+ 1)!, if j > i− 1

1, if j = i− 1

0, if j < i− 1

In this way, expanding down the last column of the corresponding determinant

yields an analogue of equation (5) for any level-decomposable set D:

(7) ΞD(n, α) =

−

n−1∑i=0

(n

i

)ΞD(n− i, α + i)LD(i, α + i) if n > 0

0 if n = 0

But what do these numbers count? In general, these do not correspond to digraphs

with free edges in the same way that they do for automata. Instead, for nonnegative

integer values of α, ΞD(n, α) counts graded digraphs on [n + α] vertices in which

30


vertices in [n] are assigned heights as normal, and the other α vertices are sinks with

no assigned heights to which any edges may be freely attached.

While ΞA(n, α) may be interpreted to count digraphs with free edges weighted by

α, for general choices of D this is not so. Instead, it’s best to count such graphs by

simply defining a new set D′ in which free edges are allowed, and to compute the cor-

responding numbers LD′(a, b). The recurrence (7) only aligns with this interpretation

in the event that LD′(a, b) = LD(a, b+ α).

31

CHAPTER 2

Other Connectivity Results

2.1. Restricted Fencings

In the early sections of this chapter, we will count digraphs with other types of

connectivity by looking at restricted fencings. That is, we will impose a condition

on the set of allowable fences and consider the sum from Theorem 1.4.1 restricted to

those fencings.

Again, this sum will cancel in two different ways from two involutions. The first

is analogous to Φ: we will look for admissible minor fences within our restriction,

choose the first such fence under the ordering ≺, and add or delete it. The second

involution is simply the restriction of Ψ to this smaller set of fencings. Since Ψ only

ever adds a fence if it is the union or intersection of two other fences, this restriction

will be well defined so long as the restricted set of fences is closed under unions and

intersections.

Unpaired from Φ is the set of major fencings in which no minor fences are admis-

sible. Usually, this will correspond (although perhaps not immediately) to the new

sort of connectivity we hope to find. Left over from Ψ is the set of gradings in which

the fences obey this restriction, which usually will correspond to some condition on

the levels. In each section, we will describe the restricted set of fences, confirm that

it is closed under unions and intersections, and explain what these two leftover sets

are and how to count them.

32

CHAPTER 2. OTHER CONNECTIVITY RESULTS

2.2. Initially Connected Digraphs

An initially connected digraph on [n] is a digraph such that every vertex is reach-

able from vertex 1. Initially connected digraphs were counted by Robinson in [7]

when D is the set of graphs without loops or multiple edges, and in [8] when D is the

set of d-regular automata.

1

2

3

45

6

7

8

Figure 2.1. An initially connected digraph on 8 vertices.

In this case, we’ll restrict our fencings to all those fences which contain the vertex

1. This is clearly closed under unions and intersections.

What is left over from the involution Φ? Such a fencing must have one major

fence, since the major fences are disjoint but must all contain 1. Furthermore, it has

no admissible minor fences containing 1, which means that the underlying digraph

is initially connected: if the set of all vertices reachable from 1 were not the entire

digraph, then it would comprise an admissible minor fence. Conversely, if there is

an admissible fence containing 1, then there cannot be a path from 1 to any vertex

outside that fence, so a digraph is included in this sum if and only if it is initially

connected. Furthermore, the weight of each such digraph is always −1, since there is

always exactly one major fence.

33


On the other hand, Ψ cancels out all but the nested fencings, and these nested

fencings have the property that vertex 1 is in every fence. This means that 1 is at

the highest level.

Let ΞTD(n) be the sum over all fencings of digraphs in Dn with this restriction, and

let CID(n) be the number of initially connected digraphs in Dn. On the one hand, we

have ΞTD(n) = −CI

D(n). But the interpretation as gradings with 1 at the highest level

also allows us to find a recurrence for ΞTD(n).

Theorem 2.2.1. Suppose D is level-decomposable. Then the sum ΞTD(n) over all

graded labeled digraphs on n vertices weighted by (−1)#levels in which vertex 1 is at

the greatest height is given by the following recurrence:

(8) ΞTD(n) =

−L(n, n)−

n−1∑i=1

(n− 1

i

)L(i, n)ΞT

D(n− i) if n > 0

0 if n = 0

Proof. The proof is almost identical to that of Equation (2). The only difference

is that we must not choose vertex 1 to be at the lowest level unless all vertices are at

the lowest level. �

Corollary 2.2.2. The number CI(n) of initially connected digraphs on n vertices

is given by the following recurrence:

(9) CI(n) =

L(n, n)−

n−1∑i=1

(n− 1

i

)L(i, n)CI(n− i) if n > 0

0 if n = 0

Proof. This follows immediately from the substitution ΞTD(n) = −CI

D(n). �

34


In the case where D is the set of d-regular automata, Equation (2.2.2) is equivalent

to one given by Robinson in [8]. In fact, his proof can be extended to our general

recurrence:

Proof. Consider the set of all digraphs in D. On the one hand, there are L(n, n)

such digraphs, because we must simply draw outgoing edges from each vertex with

no restriction on where those edges lead.

On the other hand, we can sum over the size of the set of vertices not reachable

from vertex 1. For each i, such a digraph can be uniquely determined in the following

way:

(1) First, we choose which i of the n− 1 vertices are not reachable from vertex

1. This can be done in

(n− 1

i

)ways.

(2) Next, we create an initially connected digraph on the remaining n−i vertices.

This can be done in I(n− i) ways.

(3) Finally, we draw edges from the remaining i vertices. These can lead to any

of the n vertices of the graph, so this can be done in L(i, n) ways.

All together, this says that

n−1∑i=0

(n− 1

i

)CI(n− i)L(i, n) = L(n, n).

Solving for CI(n) yields the recurrence (9). �

2.3. W -initially connected Digraphs

For a set of vertices W ⊆ V (D), we say that D is W -initially connected if every

vertex in V (D) is reachable from some vertex in W . Graded digraphs may also be

used to count W -initially connected digraphs.

35


1

23

4 5 6

7

8

Figure 2.2. A {1, 4, 5, 6}-initially connected digraph on 8 vertices.

Consider the set of all fencings in which W is contained in every fence, weighted

by (−1)#fences. We again apply analogues of Φ and Ψ to this restricted set.

Again, inclusion of W is preserved under unions and intersections, so Ψ is well-

defined. By the same reasoning as in the previous section, the analogue of Φ cancels

out all digraphs except those which are W -connected, and again gives each one weight

−1. Left over from Ψ is the set of all gradings in which all vertices in W are at the

highest level.

If D is level-decomposable then it must be label-invariant, so the number of W -

connected digraphs in D depends only on |W |. By a similar argument to Theorem

2.2.1 and Corollary 2.2.2 we have:

Theorem 2.3.1. For k ≤ n, let CID(n, k) be the number of W -initially connected

digraphs on [n] with W = [k]. Then

(10) CID(n, k) = L(n, n)−

n−k∑i=1

(n− ki

)L(i, n)CI

D(n− i, k), for n ≥ k > 0.

Note that ID(n) from Corollary 2.2.2 is equal to ID(n, 1). A similar proof to that

of Robinson can explain this recurrence, as in the previous section.

36


2.4. Initially-Finally Connected Digraphs

A digraphs D is initially-finally connected from v1 to v2 if for all v ∈ V (D),

v1 v and v v2. In [2] and later [3], Jovovic and Kilibarda count digraphs which

are initially-finally connected from some v1 to some v2. With gradings, we can count

initially-finally connected digraphs for a fixed choice of v1 and v2. In fact, we will

count an even broader class of digraphs, of which these are a particular case.

1

2

3

4

5

6

7

8

9

10

Figure 2.3. A digraph on 10 vertices which is initially-finally con-nected from 1 to 2.

A digraph D with (possibly intersecting) vertex subsets V1 and V2 is initially-

finally connected from V1 to V2 if for all v ∈ V (D), there exists v1 ∈ V1, v2 ∈ V2 so

that v1 v and v v2. Note that if V1 = {v1} and V2 = {v2}, then these are

exactly the digraphs which are initially-finally connected from v1 to v2. Also note

that if V1 = V2 = {v} for some v ∈ V (D), then the initially-finally connected graphs

from V1 to V2 are exactly the strongly connected digraphs.

Consider the restricted set of fences satisfying the condition that each fence con-

tains all elements of V1 or no elements V2. The set of allowable fences is closed under

union: if two or more fences contain no elements of V2, then neither does their union;

if some fence contains V1, then so does its union with any other set of fences. It is

37


also closed under intersection: if two or more fences contain V1, then so does their in-

tersection; if some fence contains no elements of V2, then neither does its intersection

with any other fences.

So we can apply the involutions Φ and Ψ on these fencings, and the middle sum

from Theorem 1.4.1 will reduce to a set of major fencings with no admissible minor

fences, and a set of nested fencings satisfying the fence restriction.

Let’s first consider the digraphs left over from Φ: major fencings of digraphs with

no admissible minor fences. To meet the restriction, any major fence which contains

some vertex of V2 must also contain all vertices from V1. Since the major fences

partition the vertex set, this means that V1 ∪ V2 must lie in a single major fence.

We will show that the major fence containing V1 and V2 has no admissible minor

fences if and only if it is initially-finally connected from V1 to V2.

First, we note that this major fence is initially connected from V1 if and only if it

has no admissible minor fences that include V1: if there were such a fence, then the

vertices outside of it would not be reachable from V1. Conversely, if the set of vertices

reachable from V1 is not the entire major fence, then that set is a valid minor fence

which meets the restriction.

Similarly, the major fence is finally connected to V2 if and only if it has no ad-

missible minor fences that exclude V2. For if there were such a minor fence, then its

contents cannot reach any element of V2. And if some collection of vertices V ′ cannot

reach any vertex in V2, then the set of vertices reachable from V ′ is an admissible

minor fence which excludes V2.

So the major fence containing V1 and V2 is initially-finally connected from V1 to

V2. What of the other major fences? They must be strongly connected: any subset of

those fences excludes V2, and so they must not be admissible fences. A major fence

38


has no admissible fenced subsets if and only if it strongly connected, as seen in the

proof of Theorem 1.4.1.

So any element which remains from the involution Φ is a digraph D partitioned

into several components, one of which is initially-finally connected from V1 to V2 and

the rest of which are strongly connected, weighted by (−1)#blocks of the partition. (Note

that the initially-finally connected subgraph counts as one towards this exponent,

even if it actually consists of multiple strong components.)

1

2

3

4

56

7

8

9

10

11

12

V1

V2

Figure 2.4. A typical digraph left over from Φ.

Suppose D is strongly decomposable. Then the sum over these remaining digraphs

can be expressed simply in terms of the sizes of V1 and V2, together with the sizes of

their union. Suppose |V1| = k1, |V2| = k2, and |V1 ∪ V2| = k3.

Let ΞTBD (n, k1, k2, k3) be the sum over the fencings of digraphs in D on [n] following

the restriction that each fence contains V1 or excludes V2, where V1, V2 ⊆ [n], |V1| = k1,

|V2| = k2, and |V1 ∪ V2| = k3. Furthermore, let CIFD (n, k1, k2, k3) be the number of

initially-finally connected digraphs from V1 to V2 on [n] in D, where again V1 and V2

satisfy those same conditions. Then we have

39


(11) ΞTBD (n, k1, k2, k3) =

−

n∑i=k3

(n− k3i− k3

)CIFD (i, k1, k2, k3)ΞD(n− i) if n ≥ k3

0 if n < k3

Now, the involution Ψ leaves us with all nested fencings satisfying the restriction

that each fence contains V1 or excludes V2. In terms of graded digraphs, this means

that for each i, either every vertex from V1 has height at least i, or every vertex from

V2 has height less than i. Equivalently, the height of every vertex from V2 is less than

or equal to the height of every vertex from V1.

Suppose D is level-decomposable. Now by our main result, ΞTBD (n, k1, k2, k3) also

counts the number of graded digraphs satisfying this condition, where again |V1| = k1,

|V2| = k2, |V1 ∪ V2| = k3. Note that because each vertex in V1 must be no lower than

each vertex in V2, all vertices in V1 ∩ V2 must be at the same level. Consider the

lowest level of a graded digraph with this property. It either contains not all of the

vertices from V2 (and therefore no vertices from V1), or all vertices from V2 (and any

number of vertices from V1).

In the first case, we sum over the number i of vertices at the lowest level, and

the number j of vertices from V2 at the lowest level. Note that j is at most k2 − 1.

In the second case, we sum over the number i of vertices at the lowest level, and

automatically take k2 of them to be the vertices from V2. The rest of the grading can

be filled out arbitrarily.

40


(12)

ΞTBD (n, k1, k2, k3) = −

n∑i=1

k2−1∑j=0

(k3 − k1

j

)(n− k3i− j

)LD(i, n)ΞTB

D (n−i, k1, k2−j, k3−j)

−n∑

i=k2

(n− k2i− k2

)LD(i, n)ΞD(n− i)

with the initial condition ΞTBD (n, k1, k2, k3) = 0 if any of its parameters are negative.

In the case where k1 = k2 = 1, k3 = 2, which is to say the case for initially-finally

connected digraphs from v1 to v2 for distinct v1, v2, we have:

ΞTBD (n, 1, 1, 2) = −

n−2∑i=1

(n− 2

i

)LD(i, n)ΞTB

D (n− i, 1, 1, 2)

−n∑i=1

(n− 1

i− 1

)LD(i, n)ΞD(n− i).

2.5. Finally Connected Digraphs

A digraph D is finally connected to 1 if for all v ∈ V (D), v 1. Let CFD(n) denote

the number of digraphs in D which are finally connected to 1.

A digraph on [n] is finally connected to 1 if and only if it’s initially-finally con-

nected from [n] to {1}. So this is in fact just a special case of the above result, with

k1 = k3 = n, k2 = 1. The recurrence reduces to

ΞTBD (n, n, 1, n) = −

n∑i=1

(n− 1

i− 1

)LD(i, n)ΞD(n− i).

Intuitively, such a graded digraph must contain vertex 1 in the bottom row, and

the rest of the digraph can be chosen arbitrarily. Furthermore, this recurrence counts

41


finally connected digraphs directly, since the major fence containing 1 must also

contain all of V1 = [n], and so we are counting finally connected digraphs all with

sign −1. So CFD(n) = −ΞTB

D (n, n, 1, n).

Theorem 2.5.1. Let CFD(n) denote the number of finally connected digraphs on

[n]. Then we have

CFD(n) =

n∑i=1

(n− 1

i− 1

)LD(i, n)ΞD(n− i) if n > 0

0 if n < 1.

More generally, we can count digraphs that are finally connected to W ⊆ V (D),

which is to say that for all v ∈ V (D), there exists a w ∈ W such that v w.

These are the digraphs initially-finally connected from [n] to W , and the recurrence

is similar. Let |W | = k. Then we want to compute ΞTBD (n, n, k, n), which satisfies the

following recurrence:

ΞTBD (n, n, k, n) = −

n∑i=k

(n− ki− k

)L(i, n)ΞD(n− i)

Similarly, if we let CFD(n, k) denote the number of W -finally connected digraphs

on [n] with |W | = k, then we have

CFD(n, k) =

−

n∑i=k

(n− ki− k

)LD(i, n)ΞD(n− i) if n ≥ k

0 if n < k.

In many of the cases we’re interested in, these results on finally-connected digraphs

are superfluous, since we could simply reverse the direction of all edges and end up

with a new digraph in D which is initially connected. However, some choices of D42


are not closed under such reversal, such as digraphs with outdegree restrictions. For

these sets, the results in this section are needed.

2.6. Computations

We now have all the tools we need to count strongly connected, initially connected,

finally connected, and initially-finally connected digraphs for every set D for which we

have found LD(a, b). For strongly connected digraphs, if D is strongly decomposable,

we can use Equation (2) and Theorem 1.3.1. If D is not strongly decomposable, such

as in the case of Proposition 1.9.1 when α 6= 1, there may still be hope. For example,

in this case, we can modify Theorem 1.3.1 to account for the i(n − i) missing edges

between the strong component containing 1 and the rest of the graph:

Theorem 2.6.1. Let D be a set of digraphs in which missing edges are weighted

α. Then

ΞD(n) = −n∑i=1

(n− 1

i− 1

)CSD(i)ΞD(n− i)αi(n−i).

For the case of initially-finally connected digraphs, we can use Equation (11) and

Equation (12). For finally connected digraphs, we have Theorem 2.5.1. Some results

of these computations can be found in Appendix B.

2.7. Digraphs with an Even Number of Edges

Let D be the set of digraphs with no loops or multiple edges. We can write

D = E ∪ O, where E consists of the digraphs with an even number of edges, and O

the digraphs with an odd number of edges.

Unfortunately, neither E nor O are level-decomposable: we cannot impose a re-

striction on the number of outgoing edges from each level so that independent choices

cause the total number of edges to be even or odd. In fact, E and O are not strongly

43


decomposable either: a totally cyclic digraph with an even number of edges might

contain components with an odd number of edges, and so on.

Nevertheless, strongly connected digraphs in E andO have been previously counted

in [6], and in this section we will re-derive these results using graded digraphs.

Proposition 2.7.1. Let n ≥ 2. Then ΞE(n) = ΞO(n) = ΞD(n)/2.

Proof. It suffices to show that ΞE(n) = ΞO(n), so we will find a sign-preserving

bijection between graded digraphs with an even and odd number of edges.

For a particular graded digraph, choose the lowest-numbered vertex i at the bot-

tom level, and the let j be the lowest-numbered vertex not equal to i. If there is no

edge from i to j, add one; if there is an edge from i to j, delete it. Either way, we’ve

changed the parity of the number of edges while preserving the sign of the grading.

Furthermore, this map is its own inverse, so it is a bijection. �

In the cases n = 0 and n = 1, it’s easy to see that ΞE(n) = ΞD(n), while ΞO(n) = 0.

So in summary, we have:

ΞE(n) =

1 if n = 0

−1 if n = 1

ΞD(n)/2 if n ≥ 2

(13)

ΞO(n) =

0 if n = 0 or n = 1

ΞD(n)/2 if n ≥ 2

(14)

We can use this to compute the number of strong digraphs according to the number

of edges.

44


Proposition 2.7.2. Let n ≥ 2. Then

CSE (n) =

CSD(n) + (n− 1)!

2

and

CSO(n) =

CSD(n)− (n− 1)!

2.

Proof. Consider the set D− of digraphs with no loops or multiple edges, in

which edges are weighted by −1. A digraph in D− with an even number of edges

has weight 1, while a digraph with an odd number of edges has weight −1, and so

CSD−(n) = CS

E (n)− CSO(n). Because CS

D(n) = CSE (n) + CS

O(n), it suffices to show that

CSD−(n) = CS

E (n)− CSO(n) = (n− 1)! .

By equations (13) and (14) we have:

ΞD−(n) =

1 if n = 0

−1 if n = 1

0 if n ≥ 2

So the exponential generating function for ΞD−(n) is simply 1 − x. That means

the exponential generating function for CSD−(n) is

− log(1− x) = x+x2

2+x3

3+x4

4+ · · · =

∞∑n=1

(n− 1)!xn

n!,

which completes the proof. �

2.8. A Graded Interpretation of Some Generating Function Identities

In this section, we will see how graded digraphs can lead to a shorter combinato-

rial understanding of some previously discovered identities of exponential generating

functions.

45


Both of these identities are of the form f(x) = − log(1− g(x)). Since − log(1−x)

is the exponential generating function for cycles, the “ideal” combinatorial proof of

such an identity would be a decomposition of the objects counted by f into a cyclical

arrangement of objects counted by g, but unfortunately such proofs remain elusive.

Instead, the twin roles of ΞD(n) as counting totally cyclic and graded digraphs

will allow for a different correspondence: the totally cyclic interpretation allows us

to understand what the logarithms mean, and the graded interpretation makes the

graphs easy to count. The identities, then, will reduce to an enumeration of the

relative numbers of graded digraphs satisfying certain properties, which will follow

easily from some bijections.

Let s(x) be the exponential generation function for strongly connected digraphs

with no loops or multiple edges, and tn be the number of strongly connected tourna-

ments on n vertices. In [12] and later [7] it was shown that

(15) s(x) = − log

(1−

∑2(n

2)tnxn

n!

).

We will interpret the coefficients of these generating functions in terms of graded

digraphs, and from this develop a combinatorial proof.

Note that if T is the set of all tournaments, then ΞT (n) counts totally cyclic tour-

naments on n vertices weighted by the number of components. But every tournament

has exactly one weak component, so in fact ΞT (n) = −tn.

Second, if D is the set of digraphs with no loops or multiple edges, then ΞD(x) is

exactly the inside of the logarithm in the right-hand side of Equation (15).

In other words, we want to show that

∑n

ΞD(n)xn

n!= 1 +

∑n

2(n2)ΞT (n)

xn

n!,

46


which is to say that for n ≥ 1, ΞD(n) = 2(n2)ΞT (n).

How do we count ΞD(n) in terms of graded digraphs? For each pair of vertices u

and v at the same level, we must make two choices: whether or not there is an edge

from u to v, and whether or not there is an edge from v to u. For each pair of vertices

u and v at different levels, we must make one choice: whether or not there is an edge

in the upward direction.

How do we count ΞT (n) in terms of graded digraphs? For each pair of vertices u

and v at the same level, we must make one choice: whether the edge between them

leads from u to v or from v to u. For each pair of vertices u and v at different levels,

we make no choice: there must be an edge between them, and it must point in the

upward direction.

So for each pair of vertices u and v, ΞD(n) requires making one more binary choice

than ΞT (n), so ΞD(n) = 2(n2)ΞT (n).

A similar result exists for automata. Let A be the set of d-regular automata,

and let M be the set of automata in which vertex 1 has degree d− 1 while all other

vertices have degree d. In [8] it has been shown that

sA(x) = − log(1− i(x)),

with

i(x) =∑n

nCIM(n)

xn

n!.

Again, interpreting these numbers in terms of graded digraphs yields a more

combinatorial proof of this fact. The interior of the logarithm is simply∑

ΞA(n),

while CIM

(n) = −ΞTM(n). So we want to show, for n > 0, that ΞA(n) = nΞT

M(n).

47

CHAPTER . OTHER CONNECTIVITY RESULTS

ΞA(n) counts graded automata, while nΞTM(n) counts graded d-regular automata

in which 1 is at the highest level and has degree d− 1, while some arbitrary vertex is

highlighted. We would like to show that the number of these is the same.

We modify the latter by swapping the label of 1 with that of the highlighted

vertex, then removing the highlighting. So nΞTM(n) counts graded d-regular digraph

in which some vertex at the top level only has degree d− 1.

How do we convert this almost-d-regular automaton into a d-regular automaton?

By drawing in the the missing edge. But if there are k vertices at the top level, then

there are k ways to do this.

How do we convert a d-regular automaton into an almost-d-regular automaton?

By deleting the dth edge from a vertex in the top level. Again, if there are k vertices

at the top level, then there are k ways to do this. So the number of each is the same,

which completes the proof.

48

APPENDIX A

Level Decomposability

A.1. Level Charts

In this appendix, we will formally treat the idea of level-decomposability, and

define a new object which is counted by the numbers LD(a, b).

We turn to the language of level charts. Intuitively, all we need to know in order

to build a graded digraph is how to draw the outgoing edges from the vertices at each

level. Level charts are the ways in which this can be done.

For a ≤ b, an (a, b)-level chart is a labeled digraph on [b] in which each edge

originates at a vertex in [a]. Every digraph D ∈ Gr(D) with a vertices at height i

and b − a vertices at height greater than i can be restricted to an (a, b)-level chart

Da,b in the following way:

(1) All edges not originating at height i are deleted.

(2) All vertices at height less than i are deleted.

(3) The remaining vertices are relabeled with the elements of [b] so that vertex

u has label less than that of vertex v if either:

(a) u was at a lower height than v in D, or

(b) u and v were at the same height in D, but u had a lower label than v.

Figure A.1 shows a graded digraph and its corresponding (2, 6)-level chart made

by restricting to the vertices at height greater than or equal to 2.

A graded digraph D with i levels, therefore, has i associated level charts. If we

know each of these restrictions as well as the height of each vertex in D, we can

49

CHAPTER A. LEVEL DECOMPOSABILITY

1 2

63 4 5

1 2

3

4

5

6

7

8

height 4

height 3

height 2

height 1

Figure A.1. A graded digraph on [8], and its associated (2, 6)-level chart.

recover D: relabel each level chart to match the height information, and then overlay

the edges from each chart to obtain D. An example is shown in Figure A.2.

height 3

height 2

height 11

2

3

4

5

61 2 3

4 5 6

1

2 3

1 2

1

2

3

4

5

6

height 3

height 2

height 1

Figure A.2. Recovering a graded digraph from its vertex heights andlevel charts.

This recovery is possible whenever the level charts are compatible with the given

heights. Specifically, for each level i with a vertices at that height and b vertices at the

same or greater height, there must be exactly one (a, b)-level chart. In fact, for any

50


choice of height information and level charts satisfying this compatibility condition,

a unique graded digraph can be recovered.

We say that a set of digraphs D is level-decomposable if there exist sets Da,b of

(a, b)-level charts for each b ≥ a > 0 so that each choice of compatible heights and

charts (from the Da,b) yields a graded digraph in Gr(D), and each level restriction

of a digraph in Gr(D) yields a level chart from some Da,b. In the case where we

want to count digraphs in D by weight, we also require that the level charts have a

compatible weight function so that the weight of each digraph is equal to the product

of the weights of its restrictions.

For example, the set of all digraphsD with no multiple edges is level-decomposable.

Da,b is the set of all digraphs on [b] with no multiple edges where all edges originate

at [a]. On the other hand, the set of all planar digraphs P is not level-decomposable.

To see this, consider Figure A.3. Each level chart is the restriction of some planar

digraph (for example, the chart itself), but overlaying these charts with the given

heights yields an orientation of K5, which is nonplanar.

height 4

height 3

height 2

height 11

2

3 4

5

1

2 3 4 5

1

2 3 4

1 2

3

1

Figure A.3. A compatible set of heights and planar level charts whoserecovery is nonplanar.

Whenever D is level-decomposable, we will let LD(a, b) denote the number of

(a, b)-level charts in Da,b. In the case where D is weighted, then LD(a, b) will denote

51


the sum of the weights of all level charts in Da,b. As usual, when the choice of D is

clear from context, we will suppress the subscript.

A.2. Characterization of Level-Decomposable Sets

An (a, b)-level chart is strong if its induced subgraph on [a] is strongly connected.

Lemma A.2.1. Let D be a digraph. There exists a grading of D such that every

level chart is strong.

Proof. The relation induces a preorder on the set of strong components of

D. Choose a linear order that respects this preorder by arbitrarily resolving any ties.

Grade D according to this new order: every strong component is in its own level, and

the ith strong component under the linear order is the ith level of the graph. If an

edge leads from one level to another, then that edge points in the same direction as

the relation , so it points upward. Therefore, this is a valid grading. �

We would like to characterize the sets L of level charts which can actually arise

as the union ∪a,bDa,b for some D. We say that such a set L is closed under levels

if whenever there exists an (a, b)-level chart in L with b 6= a, there also exists a

(c, b− a)-level chart in L for some c.

Clearly if L is the set of level charts for some D, then L is closed under levels:

an (a, b)-level chart for which b 6= a cannot be formed from the top level of a graded

digraph, and so the restriction of the next level up will give a (c, b− a)-level chart.

Theorem A.2.2. Let S be a set of strong level charts which is invariant under

relabeling and closed under levels. Then there exists a unique level-decomposable set

of digraphs D whose strong level charts are exactly the elements of S.

52


Proof. The construction of D is straightforward: it is the set of all digraphs

recoverable from level charts in S.

It is important to check that each element of S is in fact a level chart for some

D ∈ Gr(D), but this is given by closure: let L ∈ S be an (a, b)-level chart. If a = b

then L ∈ Gr(mD), since any (a, a)-level chart is itself a graded digraph with one

level. If a < b, then we can let L be the level chart for the first level of D, and

pick some (c, b − a)-level chart for the next level. We repeat this process until we

pick a (d, d)-level chart for some d, and then take the recovery of all of these to find

D. Therefore, closure guarantees that no level chart is superfluous, that each one

actually does arise from some graded digraph.

To see that this choice ofD is unique, supposeD andD′ are both level-decomposable

and have the same set of strong level charts, but that D ∈ D and D /∈ D′. By Lemma

A.2.1, there exists a decomposition of D into strong level charts. Because D is level-

decomposable, each of these level charts is in S. Because they are strong, they are

also level charts for D′. Finally, because D′ is also level-decomposable, their recovery

yields an element of D′, contradicting the assumption that D /∈ D′. �

This means we can define a level-decomposable set of digraphs as long as we can

characterize its strong components. So, for example, the following choices of D are

level-decomposable, although the computation of LD(a, b) may not be straightforward:

• D is the set of digraphs whose strongly connected components are planar.

• D is the set of acyclic digraphs (so the strong components are single vertices).

• D is the set of digraphs whose strong components are all cycles.

Let P be a property of digraphs, and let PS be a property of strong components

of digraphs, together with the set of edges leaving that strong component and the

vertices to which those edges lead. Say that P decomposes into PS if whenever PS is

53


true for each strong component (and its outgoing edges) it is true for P , and vice versa.

If P decomposes into a property PS, then the set of all digraphs with property P is

level-decomposable. So any property we can check locally, component-by-component,

defines a level-decomposable set of digraphs D. This includes the following, as well

as their negations:

• D contains no multiple edges.

• D contains no loops.

• D is d-regular.

• D has no free edges.

• D has no bidirected edges.

• D has no missing edges.

• D is endowed with edge labels.

54

APPENDIX B

Tables

In the follow tables, we give the number of initially connected, strongly connected,

[2]-initially connected, and initially-finally connected (from 1 to 2) digraphs on up to

10 vertices for various choices of D.

Table 1. Digraphs with no loops or multiple edges

n Initially connected Strongly connected1 1 12 2 13 32 184 2432 16065 745472 5650806 875036672 7347747767 3913822502912 35230916155688 67524560999677952 635192093896641769 4555846432005388500992 4400410978376102609280

10 1213737290478155490406694912 1190433705317814685295399296

n [2]-initially connected Initially-finally connected from 1 to 22 4 23 48 264 3072 19725 847872 6326366 935854080 7803165287 4047469805568 36339726606568 68643722102833152 645195657407398729 4592655849294724595712 4434935529068070131936

10 1218564099097810354474844160 1195091263914144226525343488

1

23

45

6

7

55

CHAPTER B. TABLES

Table 2. Orientations of simple graphs


10 2244862895215525888851 1754419666770364130730


10 2371929382692644538195 1845696848863807308348

1

23

4

5

6

7

8

910

11

56

CHAPTER B. TABLES

Table 3. 3-regular digraphs, loops and multiple edges allowed


10 119304885455190799980000 103522038947009419407720


10 132752866411807432560000 114871569069714993985680

1

2

3

4 5

57

CHAPTER B. TABLES

Table 4. 2-regular digraphs, loops and multiple edges allowed


10 32382435580110000 20769228405364320


10 44388262944333000 28030699261565280

1

2

3

4 5

58

CHAPTER B. TABLES

Table 5. 3-regular digraphs, no loops or multiple edges

n Initially connected Strongly connected4 1 15 1008 10046 948600 9382607 1159960000 11399300008 1941717303750 18979970249109 4434372562462080 4315730943843200

10 13580247461085970560 13170238269690629400

n [2]-initially connected Initially-finally connected from 1 to 24 1 15 1012 10086 958920 9485407 1179980000 11597392008 1985569136250 19411142388909 4553863621324080 4432596557129580

10 13994887606417549440 13573998757366580680

1

2

3

4

5

6

59

CHAPTER B. TABLES

Table 6. 2-regular digraphs, no loops or multiple edges

n Initially connected Strongly connected3 1 14 72 695 5832 53466 626400 5557507 89181000 773267408 16445948400 140299872309 3833008865280 3231805694040

10 1104466970557440 923218857237600

n [2]-initially connected Initially-finally connected from 1 to 23 1 14 75 725 6318 58146 698280 6222847 101488500 883659908 19002670470 162696155409 4480124944680 3788761957470

10 1302460361193600 1091394153228960

1

2

3

4

5

6

60

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