Eulerian Polynomials and Beyond

Michelle WachsUniversity of Miami

Based on joint work with John Shareshian

Binomial Coefficients and Eulerian numbers

11 1

1 2 11 3 3 1

1 4 6 4 1

11 1

1 4 11 11 11 1

1 26 66 26 1

(nj

)=(n − 1

j − 1

)+(n − 1

j

) ⟨nj

⟩= (n − j)

⟨n − 1j − 1

⟩+(j + 1)

⟨n − 1

j

⟩

• coeff’s of polynomial(t + 1)n =

∑nj=0

(nj

)t j

• coeff’s of Eulerian polynomial

An(t) =∑n−1

j=0

⟨nj

⟩t j

• rows add to 2n • rows add to n!

• subsets of {1, 2, . . . , n} ofsize j

• permutations in Sn withj descents

• Rows are palindromic and unimodal.

Binomial Coefficients and Eulerian numbers

11 1

1 2 11 3 3 1

1 4 6 4 1

11 1

1 4 11 11 11 1

1 26 66 26 1

(nj

)=(n − 1

j − 1

)+(n − 1

j

)

⟨nj

⟩= (n − j)

⟨n − 1j − 1

⟩+(j + 1)

⟨n − 1

j

⟩

• coeff’s of polynomial(t + 1)n =

∑nj=0

(nj

)t j

• coeff’s of Eulerian polynomial

An(t) =∑n−1

j=0

⟨nj

⟩t j

• rows add to 2n • rows add to n!

• subsets of {1, 2, . . . , n} ofsize j

• permutations in Sn withj descents

• Rows are palindromic and unimodal.

Binomial Coefficients and Eulerian numbers

11 1

1 2 11 3 3 1

1 4 6 4 1

11 1

1 4 11 11 11 1

1 26 66 26 1

(nj

)=(n − 1

j − 1

)+(n − 1

j

) ⟨nj

⟩= (n − j)

⟨n − 1j − 1

⟩+(j + 1)

⟨n − 1

j

⟩

• coeff’s of polynomial(t + 1)n =

∑nj=0

(nj

)t j

• coeff’s of Eulerian polynomial

An(t) =∑n−1

j=0

⟨nj

⟩t j

• rows add to 2n • rows add to n!

• subsets of {1, 2, . . . , n} ofsize j

• permutations in Sn withj descents

• Rows are palindromic and unimodal.

Binomial Coefficients and Eulerian numbers

11 1

1 2 11 3 3 1

1 4 6 4 1

11 1

1 4 11 11 11 1

1 26 66 26 1

(nj

)=(n − 1

j − 1

)+(n − 1

j

) ⟨nj

⟩= (n − j)

⟨n − 1j − 1

⟩+(j + 1)

⟨n − 1

j

⟩

• coeff’s of polynomial(t + 1)n =

∑nj=0

(nj

)t j

• coeff’s of Eulerian polynomial

An(t) =∑n−1

j=0

⟨nj

⟩t j

• rows add to 2n • rows add to n!

• subsets of {1, 2, . . . , n} ofsize j

• permutations in Sn withj descents

• Rows are palindromic and unimodal.

Binomial Coefficients and Eulerian numbers

11 1

1 2 11 3 3 1

1 4 6 4 1

11 1

1 4 11 11 11 1

1 26 66 26 1

(nj

)=(n − 1

j − 1

)+(n − 1

j

) ⟨nj

⟩= (n − j)

⟨n − 1j − 1

⟩+(j + 1)

⟨n − 1

j

⟩

• coeff’s of polynomial(t + 1)n =

∑nj=0

(nj

)t j

• coeff’s of Eulerian polynomial

An(t) =∑n−1

j=0

⟨nj

⟩t j

• rows add to 2n • rows add to n!

• subsets of {1, 2, . . . , n} ofsize j

• permutations in Sn withj descents

• Rows are palindromic and unimodal.

Binomial Coefficients and Eulerian numbers

11 1

1 2 11 3 3 1

1 4 6 4 1

11 1

1 4 11 11 11 1

1 26 66 26 1

(nj

)=(n − 1

j − 1

)+(n − 1

j

) ⟨nj

⟩= (n − j)

⟨n − 1j − 1

⟩+(j + 1)

⟨n − 1

j

⟩

• coeff’s of polynomial(t + 1)n =

∑nj=0

(nj

)t j

• coeff’s of Eulerian polynomial

An(t) =∑n−1

j=0

⟨nj

⟩t j

• rows add to 2n • rows add to n!

• subsets of {1, 2, . . . , n} ofsize j

• permutations in Sn withj descents

• Rows are palindromic and unimodal.

Binomial Coefficients and Eulerian numbers

11 1

1 2 11 3 3 1

1 4 6 4 1

11 1

1 4 11 11 11 1

1 26 66 26 1

(nj

)=(n − 1

j − 1

)+(n − 1

j

) ⟨nj

⟩= (n − j)

⟨n − 1j − 1

⟩+(j + 1)

⟨n − 1

j

⟩

• coeff’s of polynomial(t + 1)n =

∑nj=0

(nj

)t j

• coeff’s of Eulerian polynomial

An(t) =∑n−1

j=0

⟨nj

⟩t j

• rows add to 2n • rows add to n!

• subsets of {1, 2, . . . , n} ofsize j

• permutations in Sn withj descents

• Rows are palindromic and unimodal.

Eulerian polynomials - Euler’s definition

∑i≥1

t i =t

1− t

∑i≥1

it i =t

(1− t)2

∑i≥1

i2t i =t(1 + t)

(1− t)3

∑i≥1

i3t i =t(1 + 4t + t2)

(1− t)4

Euler’s triangle

11 1

1 4 11 11 11 1

1 26 66 26 1

Euler’s definition∑i≥1

int i =t An(t)

(1− t)n+1

Leonhard Euler(1707-1783)

Euler’s exponential generating function formula∑n≥0

An(t)zn

n!=

1− t

e(t−1)z − t=

(1− t)ez

etz − tez

Eulerian polynomials - Euler’s definition

∑i≥1

t i =t

1− t∑i≥1

it i =t

(1− t)2

∑i≥1

i2t i =t(1 + t)

(1− t)3

∑i≥1

i3t i =t(1 + 4t + t2)

(1− t)4

Euler’s triangle

11 1

1 4 11 11 11 1

1 26 66 26 1

Euler’s definition∑i≥1

int i =t An(t)

(1− t)n+1

Leonhard Euler(1707-1783)

Euler’s exponential generating function formula∑n≥0

An(t)zn

n!=

1− t

e(t−1)z − t=

(1− t)ez

etz − tez

Eulerian polynomials - Euler’s definition

∑i≥1

t i =t

1− t∑i≥1

it i =t

(1− t)2

∑i≥1

i2t i =t(1 + t)

(1− t)3

∑i≥1

i3t i =t(1 + 4t + t2)

(1− t)4

Euler’s triangle

11 1

1 4 11 11 11 1

1 26 66 26 1

Euler’s definition∑i≥1

int i =t An(t)

(1− t)n+1

Leonhard Euler(1707-1783)

Euler’s exponential generating function formula∑n≥0

An(t)zn

n!=

1− t

e(t−1)z − t=

(1− t)ez

etz − tez

Eulerian polynomials - Euler’s definition

∑i≥1

t i =t

1− t∑i≥1

it i =t

(1− t)2

∑i≥1

i2t i =t(1 + t)

(1− t)3

∑i≥1

i3t i =t(1 + 4t + t2)

(1− t)4

Euler’s triangle

11 1

1 4 11 11 11 1

1 26 66 26 1

Euler’s definition∑i≥1

int i =t An(t)

(1− t)n+1

Leonhard Euler(1707-1783)

Euler’s exponential generating function formula∑n≥0

An(t)zn

n!=

1− t

e(t−1)z − t=

(1− t)ez

etz − tez

Eulerian polynomials - Euler’s definition

∑i≥1

t i =t

1− t∑i≥1

it i =t

(1− t)2

∑i≥1

i2t i =t(1 + t)

(1− t)3

∑i≥1

i3t i =t(1 + 4t + t2)

(1− t)4

Euler’s triangle

11 1

1 4 11 11 11 1

1 26 66 26 1

Euler’s definition∑i≥1

int i =t An(t)

(1− t)n+1

Leonhard Euler(1707-1783)

Euler’s exponential generating function formula∑n≥0

An(t)zn

n!=

1− t

e(t−1)z − t=

(1− t)ez

etz − tez

Eulerian polynomials - Euler’s definition

∑i≥1

t i =t

1− t∑i≥1

it i =t

(1− t)2

∑i≥1

i2t i =t(1 + t)

(1− t)3

∑i≥1

i3t i =t(1 + 4t + t2)

(1− t)4

Euler’s triangle

11 1

1 4 11 11 11 1

1 26 66 26 1

Euler’s definition∑i≥1

int i =t An(t)

(1− t)n+1

Leonhard Euler(1707-1783)

Euler’s exponential generating function formula∑n≥0

An(t)zn

n!=

1− t

e(t−1)z − t

=(1− t)ez

etz − tez

Eulerian polynomials - Euler’s definition

∑i≥1

t i =t

1− t∑i≥1

it i =t

(1− t)2

∑i≥1

i2t i =t(1 + t)

(1− t)3

∑i≥1

i3t i =t(1 + 4t + t2)

(1− t)4

Euler’s triangle

11 1

1 4 11 11 11 1

1 26 66 26 1

Euler’s definition∑i≥1

int i =t An(t)

(1− t)n+1

Leonhard Euler(1707-1783)

Euler’s exponential generating function formula∑n≥0

An(t)zn

n!=

1− t

e(t−1)z − t=

(1− t)ez

etz − tez

Eulerian numbers - combinatorial interpretation

For σ ∈ Sn,

Descent set: DES(σ) := {i ∈ [n − 1] : σ(i) > σ(i + 1)}

σ = 3.25.4.1 DES(σ) = {1, 3, 4}

Define des(σ) := |DES(σ)|. So

des(32541) = 3

Excedance set: EXC(σ) := {i ∈ [n − 1] : σ(i) > i}

σ = 32541 EXC(σ) = {1, 3}

Define exc(σ) := |EXC(σ)|. So

exc(32541) = 2

Eulerian numbers - combinatorial interpretation

For σ ∈ Sn,

Descent set: DES(σ) := {i ∈ [n − 1] : σ(i) > σ(i + 1)}

σ = 3.25.4.1 DES(σ) = {1, 3, 4}

Define des(σ) := |DES(σ)|. So

des(32541) = 3

Excedance set: EXC(σ) := {i ∈ [n − 1] : σ(i) > i}

σ = 32541 EXC(σ) = {1, 3}

Define exc(σ) := |EXC(σ)|. So

exc(32541) = 2

Eulerian numbers - combinatorial interpretation

S3 des exc

123 0 0

132 1 1

213 1 1

231 1 2

312 1 1

321 2 1

∑σ∈S3

tdes(σ) = 1+4t+t2

∑σ∈S3

texc(σ) = 1+4t+t2

11 1

1 4 11 11 11 1

1 26 66 26 1

Eulerian polynomial

An(t) =n−1∑j=0

⟨nj

⟩t j =

∑σ∈Sn

tdes(σ) =∑σ∈Sn

texc(σ)

MacMahon (1905) showed equidistribution of des and exc.Carlitz and Riordin (1955) showed equals An(t).

Eulerian numbers - combinatorial interpretation

S3 des exc

123 0 0

132 1 1

213 1 1

231 1 2

312 1 1

321 2 1

∑σ∈S3

tdes(σ) = 1+4t+t2

∑σ∈S3

texc(σ) = 1+4t+t2

11 1

1 4 11 11 11 1

1 26 66 26 1

Eulerian polynomial

An(t) =n−1∑j=0

⟨nj

⟩t j =

∑σ∈Sn

tdes(σ) =∑σ∈Sn

texc(σ)

MacMahon (1905) showed equidistribution of des and exc.Carlitz and Riordin (1955) showed equals An(t).

Eulerian numbers - combinatorial interpretation

S3 des exc

123 0 0

132 1 1

213 1 1

231 1 2

312 1 1

321 2 1

∑σ∈S3

tdes(σ) = 1+4t+t2

∑σ∈S3

texc(σ) = 1+4t+t2

11 1

1 4 11 11 11 1

1 26 66 26 1

Eulerian polynomial

An(t) =n−1∑j=0

⟨nj

⟩t j =

∑σ∈Sn

tdes(σ) =∑σ∈Sn

texc(σ)

MacMahon (1905) showed equidistribution of des and exc.Carlitz and Riordin (1955) showed equals An(t).

Unimodality of Eulerian Polynomials

Probably not a new formula

An(t) =

b n+12c∑

m=1

∑k1,...,km≥2

(n

k1 − 1, k2, . . . , km

)tm−1

m∏i=1

[ki − 1]t

where[k]t := 1 + t + · · ·+ tk−1.

Sum & Product Lemma: Let A(t) and B(t) be Positive, Unimodal,Palindromic with respective centers of symmetry cA and cB . Then

A(t)B(t) is PUP with center of symmetry cA + cB .

If cA = cB then A(t) + B(t) is PUP with center of symmetrycA.

Center of symmetry:

(m − 1) +m∑i=1

ki − 2

2=

1

2(n − 1).

Unimodality of Eulerian Polynomials

Probably not a new formula

An(t) =

b n+12c∑

m=1

∑k1,...,km≥2

(n

k1 − 1, k2, . . . , km

)tm−1

m∏i=1

[ki − 1]t

where[k]t := 1 + t + · · ·+ tk−1.

Sum & Product Lemma: Let A(t) and B(t) be Positive, Unimodal,Palindromic with respective centers of symmetry cA and cB . Then

A(t)B(t) is PUP with center of symmetry cA + cB .

If cA = cB then A(t) + B(t) is PUP with center of symmetrycA.

Center of symmetry:

(m − 1) +m∑i=1

ki − 2

2=

1

2(n − 1).

Unimodality of Eulerian Polynomials

Probably not a new formula

An(t) =

b n+12c∑

m=1

∑k1,...,km≥2

(n

k1 − 1, k2, . . . , km

)tm−1

m∏i=1

[ki − 1]t

where[k]t := 1 + t + · · ·+ tk−1.

Sum & Product Lemma: Let A(t) and B(t) be Positive, Unimodal,Palindromic with respective centers of symmetry cA and cB . Then

A(t)B(t) is PUP with center of symmetry cA + cB .

If cA = cB then A(t) + B(t) is PUP with center of symmetrycA.

Center of symmetry:

(m − 1) +m∑i=1

ki − 2

2=

1

2(n − 1).

Mahonian Permutation Statistics - q-analogs

Let σ ∈ Sn.

Inversion Number:

inv(σ) := |{(i , j) : 1 ≤ i < j ≤ n, σ(i) > σ(j)}|.

inv(3142) = 3

Major Index:

maj(σ) :=∑

i∈DES(σ)

i

maj(3142) = maj(3.14.2) = 1 + 3 = 4

Major Percy Alexander MacMahon

(1854 - 1929)

Mahonian Permutation Statistics - q-analogs

S3 inv maj

123 0 0

132 1 2

213 1 1

231 2 2

312 2 1

321 3 3

∑σ∈S3

qinv(σ) =∑σ∈S3

qmaj(σ)

= 1 + 2q + 2q2 + q3

= (1 + q + q2)(1 + q)

Theorem (MacMahon 1905)∑σ∈Sn

qinv(σ) =∑σ∈Sn

qmaj(σ) = [n]q!

where [n]q := 1 + q + · · ·+ qn−1 and [n]q! := [n]q[n − 1]q · · · [1]q

Mahonian Permutation Statistics - q-analogs

S3 inv maj

123 0 0

132 1 2

213 1 1

231 2 2

312 2 1

321 3 3

∑σ∈S3

qinv(σ) =∑σ∈S3

qmaj(σ)

= 1 + 2q + 2q2 + q3

= (1 + q + q2)(1 + q)

Theorem (MacMahon 1905)∑σ∈Sn

qinv(σ) =∑σ∈Sn

qmaj(σ) = [n]q!

where [n]q := 1 + q + · · ·+ qn−1 and [n]q! := [n]q[n − 1]q · · · [1]q

Mahonian Permutation Statistics - q-analogs

S3 inv maj

123 0 0

132 1 2

213 1 1

231 2 2

312 2 1

321 3 3

∑σ∈S3

qinv(σ) =∑σ∈S3

qmaj(σ)

= 1 + 2q + 2q2 + q3

= (1 + q + q2)(1 + q)

Theorem (MacMahon 1905)∑σ∈Sn

qinv(σ) =∑σ∈Sn

qmaj(σ) = [n]q!

where [n]q := 1 + q + · · ·+ qn−1 and [n]q! := [n]q[n − 1]q · · · [1]q

Mahonian Permutation Statistics - q-analogs

S3 inv maj

123 0 0

132 1 2

213 1 1

231 2 2

312 2 1

321 3 3

∑σ∈S3

qinv(σ) =∑σ∈S3

qmaj(σ)

= 1 + 2q + 2q2 + q3

= (1 + q + q2)(1 + q)

Theorem (MacMahon 1905)∑σ∈Sn

qinv(σ) =∑σ∈Sn

qmaj(σ) = [n]q!

where [n]q := 1 + q + · · ·+ qn−1 and [n]q! := [n]q[n − 1]q · · · [1]q

q-Eulerian polynomials

Ainv,desn (q, t) :=

∑σ∈Sn

qinv(σ)tdes(σ)

Amaj,desn (q, t) :=

∑σ∈Sn

qmaj(σ)tdes(σ)

Ainv,excn (q, t) :=

∑σ∈Sn

qinv(σ)texc(σ)

Amaj,excn (q, t) :=

∑σ∈Sn

qmaj(σ)texc(σ)

q-Eulerian polynomials

Ainv,desn (q, t) :=

∑σ∈Sn

qinv(σ)tdes(σ)

Amaj,desn (q, t) :=

∑σ∈Sn

qmaj(σ)tdes(σ)

Ainv,excn (q, t) :=

∑σ∈Sn

qinv(σ)texc(σ)

Amaj,excn (q, t) :=

∑σ∈Sn

qmaj(σ)texc(σ)

Theorem (Carlitz 1954)∑i≥1

[i ]nq t i =tAmaj,des

n (q, t)∏ni=0(1− tqi )

q-Eulerian polynomials

Ainv,desn (q, t) :=

∑σ∈Sn

qinv(σ)tdes(σ)

Amaj,desn (q, t) :=

∑σ∈Sn

qmaj(σ)tdes(σ)

Ainv,excn (q, t) :=

∑σ∈Sn

qinv(σ)texc(σ)

Amaj,excn (q, t) :=

∑σ∈Sn

qmaj(σ)texc(σ)

Theorem (Carlitz 1954 MacMahon 1916)∑i≥1

[i ]nq t i =tAmaj,des

n (q, t)∏ni=0(1− tqi )

q-analogs of Euler’s exp. generating function formula

Theorem (Stanley 1976)∑n≥0

Ainv,desn (q, t)

zn

[n]q!=

1− t

Expq(z(t − 1))− t

where

Expq(z) :=∑n≥0

q(n2)zn

[n]q!

Theorem (Shareshian & MW 2006)∑n≥0

Amaj,excn (q, t)

zn

[n]q!=

(1− tq) expq(z)

expq(ztq)− tq expq(z)

where

expq(z) :=∑n≥0

zn

[n]q!

q-analogs of Euler’s exp. generating function formula

Theorem (Stanley 1976)∑n≥0

Ainv,desn (q, t)

zn

[n]q!=

1− t

Expq(z(t − 1))− t

where

Expq(z) :=∑n≥0

q(n2)zn

[n]q!

Theorem (Shareshian & MW 2006)∑n≥0

Amaj,excn (q, t)

zn

[n]q!=

(1− tq) expq(z)

expq(ztq)− tq expq(z)

where

expq(z) :=∑n≥0

zn

[n]q!

q-Eulerian polynomials and q-Eulerian numbers

Theorem (Shareshian & MW 2006)∑n≥0

Amaj,excn (q, tq−1)

zn

[n]q!=

(1− t) expq(z)

expq(zt)− t expq(z)

where

expq(z) :=∑n≥0

zn

[n]q!

q-Eulerian polynomials and q-Eulerian numbers

Theorem (Shareshian & MW 2006)∑n≥0

Amaj,excn (q, tq−1)

zn

[n]q!=

(1− t) expq(z)

expq(zt)− t expq(z)

where

expq(z) :=∑n≥0

zn

[n]q!

Specialization of (quasi)symmetric function identity∑n≥0

n∑j=0

Qn,j(x)t jzn =(1− t)H(z)

H(zt)− tH(z),

the Qn,j(x) are what we call Eulerian quasisymmetric functions(a sum of certain fundamental quasisymmetric functions)

H(z) :=∑

n≥0 hn(x)zn (complete homogeneous symmetricfunctions)

q-Eulerian polynomials and q-Eulerian numbers

Theorem (Shareshian & MW 2006)∑n≥0

Amaj,excn (q, tq−1)

zn

[n]q!=

(1− t) expq(z)

expq(zt)− t expq(z)

where

expq(z) :=∑n≥0

zn

[n]q!

From now on

An(q, t) := Amaj,excn (q, tq−1) =

∑σ∈Sn

qmaj(σ)−exc(σ)texc(σ)

and ⟨nj

⟩q

:=∑σ∈Sn

exc(σ)=j

qmaj(σ)−exc(σ)

Palindromicity and unimodality of the q-Eulerian numbers

n\j 0 1 2 3 4

1 1

2 1 1

3 1 2 + q + q2 1

4 1 3 + 2q + 3q2 + 2q3 + q4 3 + 2q + 3q2 + 2q3 + q4 1

5 1 4 + 3q + 5q2 + ... 6 + 6q + 11q2 + ... 4 + 3q + 5q2 + ... 1

Theorem (Shareshian and MW)

The q-Eulerian polynomial An(q, t) =∑n−1

t=0

⟨nj

⟩qt j is

palindromic in the sense that⟨

nj

⟩q

=⟨

nn − 1− j

⟩qfor

0 ≤ j ≤ n−12

q-unimodal in the sense that⟨

nj

⟩q−⟨

nj − 1

⟩q∈ N[q] for

1 ≤ j ≤ n−12

Palindromicity and unimodality of the q-Eulerian numbers

Proof: We use our q-analog of Euler’s exponential generatingfunction formula to prove

An(q, t) =

b n+12c∑

m=1

∑k1,...,km≥2

[n

k1 − 1, k2, . . . , km

]q

tm−1m∏i=1

[ki − 1]t ,

where [n

k1, . . . , km

]q

=[n]q!

[k1]q! · · · [km]q!

Then apply the Sum & Product Lemma.

Geometric Interpretation of the Eulerian numbers

(⟨n0

⟩,⟨

n1

⟩, . . . ,

⟨n

n − 1

⟩)is the h-vector of the type An−1 Coxeter

complex ∆n.

Stanley (1980): The h-vector of every simplicial convex polytope isunimodal (and palindromic).

This is part of the celebrated g -theorem of Billera, Lee and Stanley.

Proof idea: Let P be a d-dimensional convex polytope in Rd withinteger vertices. Let VP be the toric variety associated with P.

Danilov (1978): The h-vector of P equals

(dimH0(VP), dimH2(VP), . . . , dimH2d(VP))

(Hence⟨

nj

⟩= dimH2j(V∆n).)

Geometric Interpretation of the Eulerian numbers

(⟨n0

⟩,⟨

n1

⟩, . . . ,

⟨n

n − 1

⟩)is the h-vector of the type An−1 Coxeter

complex ∆n.

Stanley (1980): The h-vector of every simplicial convex polytope isunimodal (and palindromic).

This is part of the celebrated g -theorem of Billera, Lee and Stanley.

Proof idea: Let P be a d-dimensional convex polytope in Rd withinteger vertices. Let VP be the toric variety associated with P.

Danilov (1978): The h-vector of P equals

(dimH0(VP), dimH2(VP), . . . , dimH2d(VP))

(Hence⟨

nj

⟩= dimH2j(V∆n).)

Geometric Interpretation of the Eulerian numbers

(⟨n0

⟩,⟨

n1

⟩, . . . ,

⟨n

n − 1

⟩)is the h-vector of the type An−1 Coxeter

complex ∆n.

Stanley (1980): The h-vector of every simplicial convex polytope isunimodal (and palindromic).

This is part of the celebrated g -theorem of Billera, Lee and Stanley.

Proof idea: Let P be a d-dimensional convex polytope in Rd withinteger vertices. Let VP be the toric variety associated with P.

Danilov (1978): The h-vector of P equals

(dimH0(VP), dimH2(VP), . . . , dimH2d(VP))

(Hence⟨

nj

⟩= dimH2j(V∆n).)

Geometric Interpretation of the Eulerian numbers

Theorem (Hard Lefschetz Theorem)

Let V be a “smooth” irreducible complex projective variety of(complex) dimension m. Then for some ω ∈ H2(V) and alli = 0, . . . ,m, the map H i (V)→ H2m−i (V), given by multiplicationby ωm−i in the singular cohomology ring H∗(V), is a vector spaceisomorphism.

H2j(V)ω−→ H2(j+1)(V)

ω−→ · · · ω−→ H2m−2j(V)

=⇒ H2j(V)ω−→ H2(j+1)(V) is injective.

=⇒ dimH2j(V) ≤ dimH2(j+1)(V)

Consequently the Poincare polynomial∑m

j=0 dimH2j(V)t j isunimodal and palindromic.

Hence An(t) =∑n−1

j=0 dimH2j(V∆n)t j is unimodal and palindromic

Geometric Interpretation of the Eulerian numbers

Theorem (Hard Lefschetz Theorem)

Let V be a “smooth” irreducible complex projective variety of(complex) dimension m. Then for some ω ∈ H2(V) and alli = 0, . . . ,m, the map H i (V)→ H2m−i (V), given by multiplicationby ωm−i in the singular cohomology ring H∗(V), is a vector spaceisomorphism.

H2j(V)ω−→ H2(j+1)(V)

ω−→ · · · ω−→ H2m−2j(V)

=⇒ H2j(V)ω−→ H2(j+1)(V) is injective.

=⇒ dimH2j(V) ≤ dimH2(j+1)(V)

Consequently the Poincare polynomial∑m

j=0 dimH2j(V)t j isunimodal and palindromic.

Hence An(t) =∑n−1

j=0 dimH2j(V∆n)t j is unimodal and palindromic

Geometric Interpretation of the Eulerian numbers

Theorem (Hard Lefschetz Theorem)

Let V be a “smooth” irreducible complex projective variety of(complex) dimension m. Then for some ω ∈ H2(V) and alli = 0, . . . ,m, the map H i (V)→ H2m−i (V), given by multiplicationby ωm−i in the singular cohomology ring H∗(V), is a vector spaceisomorphism.

H2j(V)ω−→ H2(j+1)(V)

ω−→ · · · ω−→ H2m−2j(V)

=⇒ H2j(V)ω−→ H2(j+1)(V) is injective.

=⇒ dimH2j(V) ≤ dimH2(j+1)(V)

Consequently the Poincare polynomial∑m

j=0 dimH2j(V)t j isunimodal and palindromic.

Hence An(t) =∑n−1

j=0 dimH2j(V∆n)t j is unimodal and palindromic

Geometric interpretation of the q-Eulerian numbers

There is a natural action of Sn on the toric variety V∆n whichinduces a representation on each H2j(V∆n).

{Representations of Sn}ch−→ Λn

Zpsq−−→ Z[q]

ch: Frobenius characteristicΛnZ: homogeneous symmetric functions over Z of degree n

psq: stable principal specialization.

psq(f (x1, x2, . . . )) = f (1, q, q2, . . . )n∏

i=1

(1− qi )

Theorem (Shareshian & MW)⟨nj

⟩q

= psq(chH2j(V∆n))

Follows from∑n≥0

n∑j=0

Qn,j(x)t jzn =(1− t)H(z)

H(zt)− tH(z)=∑n≥0

n∑j=0

chH2j(V∆n)t jzn

Shareshian and MW Procesi and Stanley

Geometric interpretation of the q-Eulerian numbersThere is a natural action of Sn on the toric variety V∆n whichinduces a representation on each H2j(V∆n).

{Representations of Sn}ch−→ Λn

Zpsq−−→ Z[q]

ch: Frobenius characteristicΛnZ: homogeneous symmetric functions over Z of degree n

psq: stable principal specialization.

psq(f (x1, x2, . . . )) = f (1, q, q2, . . . )n∏

i=1

(1− qi )

Theorem (Shareshian & MW)⟨nj

⟩q

= psq(chH2j(V∆n))

Follows from∑n≥0

n∑j=0

Qn,j(x)t jzn =(1− t)H(z)

H(zt)− tH(z)=∑n≥0

n∑j=0

chH2j(V∆n)t jzn

Shareshian and MW Procesi and Stanley

Geometric interpretation of the q-Eulerian numbersThere is a natural action of Sn on the toric variety V∆n whichinduces a representation on each H2j(V∆n).

{Representations of Sn}ch−→ Λn

Zpsq−−→ Z[q]

ch: Frobenius characteristicΛnZ: homogeneous symmetric functions over Z of degree n

psq: stable principal specialization.

psq(f (x1, x2, . . . )) = f (1, q, q2, . . . )n∏

i=1

(1− qi )

Theorem (Shareshian & MW)⟨nj

⟩q

= psq(chH2j(V∆n))

Follows from∑n≥0

n∑j=0

Qn,j(x)t jzn =(1− t)H(z)

H(zt)− tH(z)=∑n≥0

n∑j=0

chH2j(V∆n)t jzn

Shareshian and MW Procesi and Stanley

Geometric interpretation of the q-Eulerian numbers

The hard Lefschetz map ω commutes with the action of Sn onV = V∆n . This gives Sn-module maps for j < m

2

H2j(V)ω−→ H2(j+1)(V)

ω−→ · · · ω−→ H2m−2j(V)

=⇒ H2j(V)ω−→ H2(j+1)(V) is an Sn-module injection

=⇒ chH2(j+1)(V)− chH2j(V) is Schur-positive

=⇒⟨

nj + 1

⟩q−⟨

nj

⟩q∈ N[q]

=⇒ An(q, t) is q-unimodal and palindromic.

Rawlings’ Mahonian permutation statistic (1981)

Let 1 ≤ r ≤ n. For σ ∈ Sn, set

inv<r (σ) := |{(i , j) : 1 ≤ i < j ≤ n, 0 < σ(i)− σ(j)< r}|.DES≥r (σ) := {i ∈ [n − 1] : σ(i)− σ(i + 1)≥ r}maj≥r (σ) :=

∑i∈DES≥r

i

r = 1: inv<1(σ) = 0 maj≥1(σ) = maj(σ)

r = n: inv<n(σ) = inv(σ) maj≥n(σ) = 0

Theorem (Rawlings, 1981)∑σ∈Sn

qmaj≥r (σ)+inv<r (σ) = [n]q!

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

Rawlings’ Mahonian permutation statistic (1981)

Let 1 ≤ r ≤ n. For σ ∈ Sn, set

inv<r (σ) := |{(i , j) : 1 ≤ i < j ≤ n, 0 < σ(i)− σ(j)< r}|.DES≥r (σ) := {i ∈ [n − 1] : σ(i)− σ(i + 1)≥ r}maj≥r (σ) :=

∑i∈DES≥r

i

r = 1: inv<1(σ) = 0 maj≥1(σ) = maj(σ)

r = n: inv<n(σ) = inv(σ) maj≥n(σ) = 0

Theorem (Rawlings, 1981)∑σ∈Sn

qmaj≥r (σ)+inv<r (σ) = [n]q!

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

A(1)n (q, t) =

∑σ∈Sn

qmaj(σ) = [n]q!

A(n)n (q, t) =

∑σ∈Sn

t inv(σ) = [n]t !

A(2)n (q, t) =?

The (< 2)-inversions are

635412 635412 635412

(635142)−1 = 46.25.3.1

inv<2(σ) = des(σ−1) A(2)n (1, t) =

∑σ∈Sn

tdes(σ−1) = An(t)

Theorem (Shareshian and MW)

A(2)n (q, t) = An(q, t)

Proof involves Stanley’s theory of P-partitions, Gessel’s theory ofquasisymmetric functions, our Eulerian quasisymmetric functions.

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

A(1)n (q, t) =

∑σ∈Sn

qmaj(σ) = [n]q!

A(n)n (q, t) =

∑σ∈Sn

t inv(σ) = [n]t !

A(2)n (q, t) =?

The (< 2)-inversions are

635412 635412 635412

(635142)−1 = 46.25.3.1

inv<2(σ) = des(σ−1) A(2)n (1, t) =

∑σ∈Sn

tdes(σ−1) = An(t)

Theorem (Shareshian and MW)

A(2)n (q, t) = An(q, t)

Proof involves Stanley’s theory of P-partitions, Gessel’s theory ofquasisymmetric functions, our Eulerian quasisymmetric functions.

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

A(1)n (q, t) =

∑σ∈Sn

qmaj(σ) = [n]q!

A(n)n (q, t) =

∑σ∈Sn

t inv(σ) = [n]t !

A(2)n (q, t) =?

The (< 2)-inversions are

635412 635412 635412

(635142)−1 = 46.25.3.1

inv<2(σ) = des(σ−1) A(2)n (1, t) =

∑σ∈Sn

tdes(σ−1) = An(t)

Theorem (Shareshian and MW)

A(2)n (q, t) = An(q, t)

Proof involves Stanley’s theory of P-partitions, Gessel’s theory ofquasisymmetric functions, our Eulerian quasisymmetric functions.

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

A(1)n (q, t) =

∑σ∈Sn

qmaj(σ) = [n]q!

A(n)n (q, t) =

∑σ∈Sn

t inv(σ) = [n]t !

A(2)n (q, t) =?

The (< 2)-inversions are

635412 635412 635412

(635142)−1 = 46.25.3.1

inv<2(σ) = des(σ−1) A(2)n (1, t) =

∑σ∈Sn

tdes(σ−1) = An(t)

Theorem (Shareshian and MW)

A(2)n (q, t) = An(q, t)

Proof involves Stanley’s theory of P-partitions, Gessel’s theory ofquasisymmetric functions, our Eulerian quasisymmetric functions.

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

A(1)n (q, t) =

∑σ∈Sn

qmaj(σ) = [n]q!

A(n)n (q, t) =

∑σ∈Sn

t inv(σ) = [n]t !

A(2)n (q, t) =?

The (< 2)-inversions are

635412 635412 635412

(635142)−1 = 46.25.3.1

inv<2(σ) = des(σ−1) A(2)n (1, t) =

∑σ∈Sn

tdes(σ−1) = An(t)

Theorem (Shareshian and MW)

A(2)n (q, t) = An(q, t)

Proof involves Stanley’s theory of P-partitions, Gessel’s theory ofquasisymmetric functions, our Eulerian quasisymmetric functions.

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

So A(r)n (1, t) is a generalized Eulerian polynomial and A

(r)n (q, t) is

a generalized q-Eulerian polynomial.

r A(r)n (q, t)

1 [n]q!

2

b n+12c∑

m=1

∑k1,...,km≥2

[n

k1 − 1, k2, . . . , km

]q

tm−1m∏i=1

[ki − 1]t

... ???

n − 2 [n]t [n − 3]t ![n − 3]2t + [n]q tn−3[n − 4]t ![n − 2]t([n − 3]t + [2]t [n − 4]t)

+

[n

n − 2, 2

]q

t3n−10[n − 4]![n − 2]t [2]t

n − 1 [n]t [n − 2]t ![n − 2]t + [n]qtn−2[n − 2]t !

n [n]t !

Conjecture (Shareshian and MW)

A(r)n (q, t) is q-unimodal (and palindromic).

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

So A(r)n (1, t) is a generalized Eulerian polynomial and A

(r)n (q, t) is

a generalized q-Eulerian polynomial.

r A(r)n (q, t)

1 [n]q!

2

b n+12c∑

m=1

∑k1,...,km≥2

[n

k1 − 1, k2, . . . , km

]q

tm−1m∏i=1

[ki − 1]t

... ???

n − 2 [n]t [n − 3]t ![n − 3]2t + [n]q tn−3[n − 4]t ![n − 2]t([n − 3]t + [2]t [n − 4]t)

+

[n

n − 2, 2

]q

t3n−10[n − 4]![n − 2]t [2]t

n − 1 [n]t [n − 2]t ![n − 2]t + [n]qtn−2[n − 2]t !

n [n]t !

Conjecture (Shareshian and MW)

A(r)n (q, t) is q-unimodal (and palindromic).

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

Exercise (Stanley EC1, 1.50 f): Prove that A(r)n (1, t) is palindromic

and unimodal.Solution:

Theorem (De Mari and Shayman - 1988)

Let Hn,r be the type An−1 regular semisimple Hessenberg varietyof degree r . Then

A(r)n (1, t) =

d(n,r)∑j=0

dimH2j(Hn,r )t j

Consequently by the hard Lefschetz theorem, A(r)n (1, t) is

palindromic and unimodal.

Stanley: Is there a more elementary proof of unimodality?Shareshian and MW: Is there a q-analog of this result?(Hn,2 is the toric variety V∆n)

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

Exercise (Stanley EC1, 1.50 f): Prove that A(r)n (1, t) is palindromic

and unimodal.Solution:

Theorem (De Mari and Shayman - 1988)

Let Hn,r be the type An−1 regular semisimple Hessenberg varietyof degree r . Then

A(r)n (1, t) =

d(n,r)∑j=0

dimH2j(Hn,r )t j

Consequently by the hard Lefschetz theorem, A(r)n (1, t) is

palindromic and unimodal.

Stanley: Is there a more elementary proof of unimodality?

Shareshian and MW: Is there a q-analog of this result?(Hn,2 is the toric variety V∆n)

A(r)n (q, t) :=

∑σ∈Sn

qmaj≥r (σ)t inv<r (σ)

Exercise (Stanley EC1, 1.50 f): Prove that A(r)n (1, t) is palindromic

and unimodal.Solution:

Theorem (De Mari and Shayman - 1988)

Let Hn,r be the type An−1 regular semisimple Hessenberg varietyof degree r . Then

A(r)n (1, t) =

d(n,r)∑j=0

dimH2j(Hn,r )t j

Consequently by the hard Lefschetz theorem, A(r)n (1, t) is

palindromic and unimodal.

Stanley: Is there a more elementary proof of unimodality?Shareshian and MW: Is there a q-analog of this result?(Hn,2 is the toric variety V∆n)

Hessenberg Varieties (De Mari and Shayman - 1988)

Let Fn be the set of all flags

F : V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn

where dimVi = i . Fix X ∈ GLn(C) with n distinct eigenvalues.

The type A regular semisimple Hessenberg variety of degree r is

Hn,r := {F ∈ Fn | XVi ⊆ Vi+r−1 for all i}

For q-unimodality we want an action of Sn on Hn,r .

The Representation

Tymoczko (2008) used a theory of Goresky, Kottwitz andMacPherson (GKM theory) to define a representation of Sn

on H2j(Hn,r ).MacPherson & Tymoczko show that the hard Lefschetz mapcommutes with the action of Sn on H2j(Hn,r ) .

Conjecture (Shareshian and MW)

A(r)n (q, t) =

d(n,r)∑j=0

psq(chH2j(Hn,r ))t j

Consequently A(r)n (q, t) is q-unimodal (and palindromic).

The r = 2 case is the toric variety case.

A(2)n (q, t) = An(q, t) =

n−1∑j=0

psq(chH2j(V∆n))t j =n−1∑j=0

psq(chH2j(Hn,2))t j

Also true for r = 1, n − 2, n − 1, n

The Representation

Tymoczko (2008) used a theory of Goresky, Kottwitz andMacPherson (GKM theory) to define a representation of Sn

on H2j(Hn,r ).MacPherson & Tymoczko show that the hard Lefschetz mapcommutes with the action of Sn on H2j(Hn,r ) .

Conjecture (Shareshian and MW)

A(r)n (q, t) =

d(n,r)∑j=0

psq(chH2j(Hn,r ))t j

Consequently A(r)n (q, t) is q-unimodal (and palindromic).

The r = 2 case is the toric variety case.

A(2)n (q, t) = An(q, t) =

n−1∑j=0

psq(chH2j(V∆n))t j =n−1∑j=0

psq(chH2j(Hn,2))t j

Also true for r = 1, n − 2, n − 1, n

Chromatic symmetric functions

1

23

4

5

67 8

Wednesday, December 7, 2011

1

23

4

5

67 8

77

15 15

23

23

23

35

23

23

Let V (G ) = {1, 2, . . . , n}. Let C (G ) be set of proper colorings ofG , where a proper coloring is a map c : V (G )→ P such thatc(i) 6= c(j) if {i , j} ∈ E (G ).

Chromatic symmetric function (Stanley, 1995)

XG (x) :=∑

c∈C(G)

xc(1)xc(2) . . . xc(n)

XG (1, 1, . . . , 1︸ ︷︷ ︸, 0, 0, . . . ) = χG (m)

m

Chromatic symmetric functions

1

23

4

5

67 8

1

23

4

5

67 8

77

15 15

23

23

23

35

23

23

Let V (G ) = {1, 2, . . . , n}. Let C (G ) be set of proper colorings ofG , where a proper coloring is a map c : V (G )→ P such thatc(i) 6= c(j) if {i , j} ∈ E (G ).

Chromatic symmetric function (Stanley, 1995)

XG (x) :=∑

c∈C(G)

xc(1)xc(2) . . . xc(n)

XG (1, 1, . . . , 1︸ ︷︷ ︸, 0, 0, . . . ) = χG (m)

m

Chromatic symmetric functions

1

23

4

5

67 8

77

15 15

23

23

23

35

23

23

Let V (G ) = {1, 2, . . . , n}. Let C (G ) be set of proper colorings ofG , where a proper coloring is a map c : V (G )→ P such thatc(i) 6= c(j) if {i , j} ∈ E (G ).

Chromatic symmetric function (Stanley, 1995)

XG (x) :=∑

c∈C(G)

xc(1)xc(2) . . . xc(n)

XG (1, 1, . . . , 1︸ ︷︷ ︸, 0, 0, . . . ) = χG (m)

m

Chromatic symmetric functions

1

23

4

5

67 8

77

15 15

23

23

23

35

23

23

Let V (G ) = {1, 2, . . . , n}. Let C (G ) be set of proper colorings ofG , where a proper coloring is a map c : V (G )→ P such thatc(i) 6= c(j) if {i , j} ∈ E (G ).

Chromatic symmetric function (Stanley, 1995)

XG (x) :=∑

c∈C(G)

xc(1)xc(2) . . . xc(n)

XG (1, 1, . . . , 1︸ ︷︷ ︸, 0, 0, . . . ) = χG (m)

m

Chromatic symmetric functions

1

23

4

5

67 8

77

15 15

23

23

23

35

23

23

Let V (G ) = {1, 2, . . . , n}. Let C (G ) be set of proper colorings ofG , where a proper coloring is a map c : V (G )→ P such thatc(i) 6= c(j) if {i , j} ∈ E (G ).

Chromatic symmetric function (Stanley, 1995)

XG (x) :=∑

c∈C(G)

xc(1)xc(2) . . . xc(n)

XG (1, 1, . . . , 1︸ ︷︷ ︸, 0, 0, . . . ) = χG (m)

m

Chromatic quasisymmetric function

1

23

4

5

67 8

77

15 15

23

23

23

35

23

23

Chromatic quasisymmetric function (Shareshian and MW)

XG (x, t) :=∑

c∈C(G)

tdes(c)xc(1)xc(2) . . . xc(n)

where

des(c) := |{{i , j} ∈ E (G ) : i < j and c(i) > c(j)}|.

Chromatic quasisymmetric function

eλ(x1, x2, . . . ) := elementary symmetric function indexed bypartition λ

Fµ(x1, x2, . . . ) := fundamental quasisymmetric functionindexed by composition µ

1 2 3G =

XG (x, t) = e3 + (e3 + e2,1)t + e3t2

1 3 2G =

XG (x, t) = (e3 + F1,2) + 2e3t + (e3 + F2,1)t2

Formulae

Let Gn,r be the graph with vertex set {1, 2, . . . , n} and edge set{{i , j} | 0 < |j − i | < r}.

r XGn,r (x, t)

1 en1

2

b n+12c∑

m=1

∑k1,...,km≥2∑

ki=n+1

ek1−1,k2,...,km tm−1m∏i=1

[ki − 1]t

... ???

n − 2 en[n]t [n − 3]t ![n − 3]2t + en−1,1 tn−3[n − 4]t ![n − 2]t([n − 3]t + [2]t [n − 4]t)

+en−2,2t3n−10[n − 4]![n − 2]t [2]t

n − 1 en[n]t [n − 2]t ![n − 2]t + en−1,1tn−2[n − 2]t !

n en[n]t !

Theorem (Shareshian and MW)

A(r)n (q, t) = psq(ωXGn,r (x, t))

Formulae

Let Gn,r be the graph with vertex set {1, 2, . . . , n} and edge set{{i , j} | 0 < |j − i | < r}.

r XGn,r (x, t)

1 en1

2

b n+12c∑

m=1

∑k1,...,km≥2∑

ki=n+1

ek1−1,k2,...,km tm−1m∏i=1

[ki − 1]t

... ???

n − 2 en[n]t [n − 3]t ![n − 3]2t + en−1,1 tn−3[n − 4]t ![n − 2]t([n − 3]t + [2]t [n − 4]t)

+en−2,2t3n−10[n − 4]![n − 2]t [2]t

n − 1 en[n]t [n − 2]t ![n − 2]t + en−1,1tn−2[n − 2]t !

n en[n]t !

Theorem (Shareshian and MW)

A(r)n (q, t) = psq(ωXGn,r (x, t))

The Main Conjecture

Conjecture (Shareshian and MW)

Let G be the incomparability graph of a natural unit interval order.Then

ωXG (x, t) =

d(n,r)∑j=0

chH2j(HG )t j

where HG is the Hessenberg variety associated with G .

⇒ q-unimodality of A(r)n (q, t) (set G = Gn,r )

⇒ Schur-positivity and Schur-unimodality of XG (x, t)

Theorem (Shareshian and MW - t-analog of Gasharov)

Let G be the incomparability graph of a natural unit interval orderP. Then

XG (x, t) =∑T∈TP

t invG (T )sλ(T ),

where TP is the set of P-tableaux, invG is an inversion statistic ontableaux, and λ(T ) is the shape of T .

The Main Conjecture

Conjecture (Shareshian and MW)

Let G be the incomparability graph of a natural unit interval order.Then

ωXG (x, t) =

d(n,r)∑j=0

chH2j(HG )t j

where HG is the Hessenberg variety associated with G .

⇒ q-unimodality of A(r)n (q, t) (set G = Gn,r )

⇒ Schur-positivity and Schur-unimodality of XG (x, t)

Theorem (Shareshian and MW - t-analog of Gasharov)

Let G be the incomparability graph of a natural unit interval orderP. Then

XG (x, t) =∑T∈TP

t invG (T )sλ(T ),

where TP is the set of P-tableaux, invG is an inversion statistic ontableaux, and λ(T ) is the shape of T .

The Main Conjecture

Conjecture (Shareshian and MW)

Let G be the incomparability graph of a natural unit interval order.Then

ωXG (x, t) =

d(n,r)∑j=0

chH2j(HG )t j

where HG is the Hessenberg variety associated with G .

⇒ q-unimodality of A(r)n (q, t) (set G = Gn,r )

⇒ Schur-positivity and Schur-unimodality of XG (x, t)

Theorem (Shareshian and MW - t-analog of Gasharov)

Let G be the incomparability graph of a natural unit interval orderP. Then

XG (x, t) =∑T∈TP

t invG (T )sλ(T ),

where TP is the set of P-tableaux, invG is an inversion statistic ontableaux, and λ(T ) is the shape of T .

Refinement of Stanley-Stembridge Conjecture

Conjecture (Shareshian and MW)

Let G be the incomparability graph of a natural unit interval order.Then XG (x, t) is e-positive and e-unimodal.

Let Gn,r be the graph with vertex set {1, 2, . . . , n} and edge set{{i , j} | 0 < |j − i | < r}.

r XGn,r

1 e1n

2

b n+12c∑

m=1

∑k1,...,km≥2∑

ki=n+1

ek1−1,k2,...,km tm−1m∏i=1

[ki − 1]t

... ???

n − 2 en[n]t [n − 3]t ![n − 3]2t + en−1,1 tn−3[n − 4]t ![n − 2]t([n − 3]t + [2]t [n − 4]t)

+en−2,2t3n−10[n − 4]![n − 2]t [2]t

n − 1 en[n]t [n − 2]t ![n − 2]t + en−1,1tn−2[n − 2]t !

n en[n]t !