Generalized Inversion Sequences

Carla D. Savage

Department of Computer ScienceNorth Carolina State University

CanaDAM 2013Memorial University of Newfoundland, June 10, 2013

Permutations and Descents

Sn: set of permutations π : {1,2, . . . ,n} → {1,2, . . . ,n}

Desπ: {i ∈ {1, . . .n − 1} | π(i) > π(i + 1)} (descents)

desπ: |Desπ|, the number of descents.

π ∈ S3 Desπ desπ1 2 3 { } 01 3 2 {2} 12 1 3 {1} 12 3 1 {2} 13 1 2 {1} 13 2 1 {1,2} 2

Descent polynomial:

En(x) =∑π∈Sn

xdesπ

E3(x) = 1 + 4x + x2

Eulerian polynomials: En(x)

Permutations and Descents

Sn: set of permutations π : {1,2, . . . ,n} → {1,2, . . . ,n}

Desπ: {i ∈ {1, . . .n − 1} | π(i) > π(i + 1)} (descents)

desπ: |Desπ|, the number of descents.

π ∈ S3 Desπ desπ1 2 3 { } 01 3 2 {2} 12 1 3 {1} 12 3 1 {2} 13 1 2 {1} 13 2 1 {1,2} 2

Descent polynomial:

En(x) =∑π∈Sn

xdesπ

E3(x) = 1 + 4x + x2

Eulerian polynomials: En(x)

Permutations and Descents

Sn: set of permutations π : {1,2, . . . ,n} → {1,2, . . . ,n}

Desπ: {i ∈ {1, . . .n − 1} | π(i) > π(i + 1)} (descents)

desπ: |Desπ|, the number of descents.

π ∈ S3 Desπ desπ1 2 3 { } 01 3 2 {2} 12 1 3 {1} 12 3 1 {2} 13 1 2 {1} 13 2 1 {1,2} 2

Descent polynomial:

En(x) =∑π∈Sn

xdesπ

E3(x) = 1 + 4x + x2

Eulerian polynomials: En(x)

The Eulerian polynomials, En(x)

En(x) =∑π∈Sn

xdesπ

∑t≥0

(t + 1)nx t =En(x)

(1− x)n+1

∑n≥0

En(x)zn

n!=

(1− x)

ez(x−1) − x

Inversion Sequences

In = {(e1, . . .en) ∈ Zn | 0 ≤ ei < i}

Encode permutations as inversion sequences φ : Sn → Inφ(π) = (e1, . . . ,en), where

ej = |{i | i < j and π(i) > π(j)}| .

Example:φ(4 3 6 5 1 2) = (0,1,0,1,4,4).

Claim: φ is a bijection with Desπ = Ascφ(π). What is “Asc ”?

Inversion Sequences

In = {(e1, . . .en) ∈ Zn | 0 ≤ ei < i}

Encode permutations as inversion sequences φ : Sn → Inφ(π) = (e1, . . . ,en), where

ej = |{i | i < j and π(i) > π(j)}| .

Example:φ(4 3 6 5 1 2) = (0,1,0,1,4,4).

Claim: φ is a bijection with Desπ = Ascφ(π).

What is “Asc ”?

Inversion Sequences

In = {(e1, . . .en) ∈ Zn | 0 ≤ ei < i}

Encode permutations as inversion sequences φ : Sn → Inφ(π) = (e1, . . . ,en), where

ej = |{i | i < j and π(i) > π(j)}| .

Example:φ(4 3 6 5 1 2) = (0,1,0,1,4,4).

Claim: φ is a bijection with Desπ = Ascφ(π). What is “Asc ”?

Ascents

What is an “ascent” in an inversion sequence?

ei < ei+1?

orei

i<

ei+1

i + 1?

Lemma

If 0 ≤ ej < j for all j ≤ n, then for 1 ≤ i < n,

ei < ei+1 iffei

i<

ei+1

i + 1.

Ascents

What is an “ascent” in an inversion sequence?

ei < ei+1? orei

i<

ei+1

i + 1?

Lemma

If 0 ≤ ej < j for all j ≤ n, then for 1 ≤ i < n,

ei < ei+1 iffei

i<

ei+1

i + 1.

Ascents

What is an “ascent” in an inversion sequence?

ei < ei+1? orei

i<

ei+1

i + 1?

Lemma

If 0 ≤ ej < j for all j ≤ n, then for 1 ≤ i < n,

ei < ei+1 iffei

i<

ei+1

i + 1.

View inversion sequences as lattice pointsin a (half-open) 1 × 2 × · · · × n box

View ascent constraints as hyperplane constraints:

0 < e1 andei

i<

ei+1

i + 1, 1 ≤ i < n

π ∈ S3 e ∈ In Asc e1 2 3 (0,0,0) { }1 3 2 (0,0,1) {2}2 1 3 (0,1,0) {1}2 3 1 (0,0,2) {2}3 1 2 (0,1,1) {1}3 2 1 (0,1,2) {1,2}

s-inversion sequencesFor any sequence s = (s1, s2, . . . , sn) of positive integers:

I(s)n = {(e1, . . . ,en) ∈ Zn | 0 ≤ ei < si for 1 ≤ i ≤ n} .

(lattice points in a half-open s1 × s2 × · · · × sn box)

∣∣∣I(s)n

∣∣∣ = s1s2 · · · sn

s-inversion sequencesFor any sequence s = (s1, s2, . . . , sn) of positive integers:

I(s)n = {(e1, . . . ,en) ∈ Zn | 0 ≤ ei < si for 1 ≤ i ≤ n} .

%pause

An ascent of e is a position i :1 ≤ i < n and

ei

si<

ei+1

si+1.

If e1 > 0 then 0 is an ascent.

s-inversion sequencesFor any sequence s = (s1, s2, . . . , sn) of positive integers:

I(s)n = {(e1, . . . ,en) ∈ Zn | 0 ≤ ei < si for 1 ≤ i ≤ n} .

%pause

Example: (2,4,5) ∈ I(3,5,7)n

Asc e = {0,1}

2 6∈ Asc e since 4/5 6< 5/7

Ascent polynomials of s-inversion sequences

A(s)n (x) =

∑e∈I(s)

n

xasc e

(2,4)-inversion sequences

A(2,4)n (x) = 1 + 6x + x2

Ascent sets:{ } yellow dot{0} blue square{1} red diamond{0,1} black dot

(3,5)-inversion sequences

A(3,5)n (x) = 1 + 10x + 4x2

Ascent polynomials of s-inversion sequences

A(s)n (x) =

∑e∈I(s)

n

xasc e

(2,4)-inversion sequences

A(2,4)n (x) = 1 + 6x + x2

Ascent sets:{ } yellow dot{0} blue square{1} red diamond{0,1} black dot

(3,5)-inversion sequences

A(3,5)n (x) = 1 + 10x + 4x2

Call A(s)n (x) the s-Eulerian polynomial since,

when s = (1,2, . . . ,n),

A(s)n (x) =

∑e∈I(s)

n

x asc e =∑π∈Sn

xdesπ = En(x),

the Eulerian polynomial.

(Recall bijection φ : Sn → In with Desπ = Ascφ(π))

Why s-inversion sequences?

• Natural model for combinatorial structures• Can prove general properties of the s-Eulerian polynomials• Surprising results follow using Ehrhart theory• Can be encoded as lecture hall partitions• Lead to a natural refinement of the s-Eulerian polynomials• Help answer questions about lecture hall partitions

sequence s s-Eulerian polynomial

(1,2,3,4,5,6) 1 + 57 x + 302 x2 + 302 x3 + 57 x4 + x5

(2,4,6,8,10) 1 + 237 x + 1682 x2 + 1682 x3 + 237 x4 + x5

(6,5,4,3,2,1) 1 + 57 x + 302 x2 + 302 x3 + 57 x4 + x5

(1,1,3,2,5,3) 1 + 20 x + 48 x2 + 20 x3 + x4

(1,3,5,7,9,11) 1 + 358 x + 3580 x2 + 5168 x3 + 1328 x4 + 32 x5.

(7,2,3,5,4,6) 1 + 71 x + 948 x2 + 2450 x3 + 1411 x4 + 159 x5

1.

Signed permutationsand

(2,4,6, . . . ,2n)-inversion sequences

∣∣∣I(2,4,6,...,2n)n

∣∣∣ = 2nn!

Signed Permutations Bn

Bn = {(σ1, . . . , σn) | ∃π ∈ Sn, ∀i σi = ±π(i)}

π ∈ Bn Desπ( 1, 2) { }(-1, 2) {0}( 1,-2) {1}(-1,-2) {0,1}( 2, 1) {1}(-2, 1) {0}( 2,-1) {1}(-2,-1) {0}

descent polynomial:

1 + 6x + x2

Desσ = {i ∈ {0, . . . ,n−1} | σi > σi+1},

with the convention that σ0 = 0.

Signed Permutations Bn

Bn = {(σ1, . . . , σn) | ∃π ∈ Sn, ∀i σi = ±π(i)}

π ∈ Bn Desπ( 1, 2) { }(-1, 2) {0}( 1,-2) {1}(-1,-2) {0,1}( 2, 1) {1}(-2, 1) {0}( 2,-1) {1}(-2,-1) {0}

descent polynomial:

1 + 6x + x2

Ascent sets:{ } yellow dot{0} blue square{1} red diamond{0,1} black dot

(2,4)-inversion sequences

ascent polynomial:

1 + 6x + x2

Theorem (Pensyl, S 2012/13)

∑σ∈Bn

xdesσ = A(2,4,...,2n)n (x).

Proof.

There is a bijection Θ : Bn → I(2,4,...,2n)n satisfying

Desσ = Asc Θ(σ).

Theorem (Pensyl, S 2012/13)

∑σ∈Bn

xdesσ = A(2,4,...,2n)n (x).

Proof.

There is a bijection Θ : Bn → I(2,4,...,2n)n satisfying

Desσ = Asc Θ(σ).

2.

(s1, s2, . . . , sn)-inversion sequencesvs.

(sn, sn−1, . . . , s1)-inversion sequences

Example from table:

A(1,2,3,4,5,6)6 (x) = 1 + 57 x + 302 x2 + 302 x3 + 57 x4 + x5

= A(6,5,4,3,2,1)6 (x)

Theorem (S, Schuster 2012; Liu, Stanley 2012)

For any sequence (s1, s2, . . . , sn) of positive integers,

A(s1,s2,...,sn)n (x) = A(sn,sn−1,...,s1)

n (x).

Reversing s preserves the ascent polynomial

(3,5)-inversion sequences

1 + 10 x + 4 x2

Ascent sets:{ } yellow dot{0} blue square{1} red diamond{0,1} black dot

(5,3)-inversion sequences

1 + 10 x + 4 x2

but not necessarily the partition into ascent sets

3.Roots of s-Eulerian polynomials

Example from table:

A(7,2,3,5,4,6)6 (x) = 1+71 x +948 x2 +2450 x3 +1411 x4 +159 x5

Roots in the intervals:

[[−19/1024,−9/512], [−77/1024,−19/256],

[−423/1024,−211/512], [−1701/1024,−425/256],

[−3435/512,−6869/1024]]

Theorem (S, Visontai 2012)

For every sequence s of positive integers, A(s)n (x) has all real

roots.

Corollary (Frobenius 1910; Brenti 1994)

The descent polynomials for Coxeter groups of types A and Bhave all real roots.

New: ([S, Visontai 2013]) Method can be adapted to type D.

Theorem (S, Visontai 2012)

For every sequence s of positive integers, A(s)n (x) has all real

roots.

Corollary (Frobenius 1910; Brenti 1994)

The descent polynomials for Coxeter groups of types A and Bhave all real roots.

New: ([S, Visontai 2013]) Method can be adapted to type D.

Theorem (S, Visontai 2012)

For every sequence s of positive integers, A(s)n (x) has all real

roots.

Corollary (Frobenius 1910; Brenti 1994)

The descent polynomials for Coxeter groups of types A and Bhave all real roots.

New: ([S, Visontai 2013]) Method can be adapted to type D.

Theorem (S, Visontai 2012)

For every sequence s of positive integers, A(s)n (x) has all real

roots.

Corollary

For any s, the sequence of coefficents of the s-Eulerianpolynomial is unimodal and log-concave.

Example A(7,2,3,5,4,6)6 (x):

1, 71, 948, 2450, 1411, 159, 1

Theorem (S, Visontai 2012)

For every sequence s of positive integers, A(s)n (x) has all real

roots.

Corollary

For any s, the sequence of coefficents of the s-Eulerianpolynomial is unimodal and log-concave.

Example A(7,2,3,5,4,6)6 (x):

1, 71, 948, 2450, 1411, 159, 1

4.

Lecture Hall Polytopesand

Ehrhart Theory

Lecture hall polytopess-lecture hall polytope:

P(s)n =

{λ ∈ Rn

∣∣∣ 0 ≤ λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn≤ 1

}.

P(3,5)2

Lecture hall polytopess-lecture hall polytope:

P(s)n =

{λ ∈ Rn

∣∣∣ 0 ≤ λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn≤ 1

}.

P(3,5)2

t-th dilation of P(s)n :

tP(s)n = {tλ | λ ∈ P(s)

n }

Ehrhart polynomial of P(s)n :

i(P(s)n , t) = |tP(s)

n ∩ Zn|.

Lecture hall polytopess-lecture hall polytope:

P(s)n =

{λ ∈ Rn

∣∣∣ 0 ≤ λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn≤ 1

}.

P(3,5)2

t-th dilation of P(s)n :

tP(s)n = {tλ | λ ∈ P(s)

n }

Ehrhart polynomial of P(s)n :

i(P(s)n , t) = |tP(s)

n ∩ Zn|.

Lecture hall polytopess-lecture hall polytope:

P(s)n =

{λ ∈ Rn

∣∣∣ 0 ≤ λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn≤ 1

}.

P(3,5)2

t-th dilation of P(s)n :

tP(s)n = {tλ | λ ∈ P(s)

n }

Ehrhart polynomial of P(s)n :

i(P(s)n , t) = |tP(s)

n ∩ Zn|.

Connection between lecture hall polytopes andinversion sequences

Theorem (S, Schuster 2012)

For any sequence s of positive integers,

∑t≥0

i(P(s)n , t) x t =

∑e∈I(s)

nx asc (e)

(1− x)n+1 .

5.

The sequences

s = (1, 1, 3, 2, 5, 3, 7, 4, . . .)

and

s = (1, 4, 3, 8, 5, 12, 7, 16, . . .)

Note:∣∣∣I(1,1,3,2,5,3,...,2n−1,n)2n

∣∣∣ = n!(1 · 3 · 5 · · · · · 2n − 1) =(2n)!

2n

Theorem (S, Visontai 2012)

A(1,1,3,2,...,2n−1,n)2n is the descent polynomial for permutations of

the multiset {1,1,2,2, . . . ,n,n}.

Bijective proof?

Conjecture (S, Visontai 2012)

A(1,4,3,8,...,2n−1,4n)2n is the descent polynomial for the signed

permutations of {1,1,2,2, . . . ,n,n}.

Theorem (S, Visontai 2012)

A(1,1,3,2,...,2n−1,n)2n is the descent polynomial for permutations of

the multiset {1,1,2,2, . . . ,n,n}.

Bijective proof?

Conjecture (S, Visontai 2012)

A(1,4,3,8,...,2n−1,4n)2n is the descent polynomial for the signed

permutations of {1,1,2,2, . . . ,n,n}.

Theorem (S, Visontai 2012)

A(1,1,3,2,...,2n−1,n)2n is the descent polynomial for permutations of

the multiset {1,1,2,2, . . . ,n,n}.

Bijective proof?

Conjecture (S, Visontai 2012)

A(1,4,3,8,...,2n−1,4n)2n is the descent polynomial for the signed

permutations of {1,1,2,2, . . . ,n,n}.

6.

The sequences s = (1, k + 1,2k + 1,3k + 1, . . .)

k = 1 : (1, 2, 3, 4, 5, . . .)k = 2 : (1, 3, 5, 7, 9, . . .)

Let:

In,k = I (1,k+1,2k+1,...,(n−1)k+1)n

An,k (x) = A(1,k+1,2k+1,...,(n−1)k+1)n (x)

Recall An,1(x) = En(x).

6.

The sequences s = (1, k + 1,2k + 1,3k + 1, . . .)

k = 1 : (1, 2, 3, 4, 5, . . .)k = 2 : (1, 3, 5, 7, 9, . . .)

Let:

In,k = I (1,k+1,2k+1,...,(n−1)k+1)n

An,k (x) = A(1,k+1,2k+1,...,(n−1)k+1)n (x)

Recall An,1(x) = En(x).

The 1/k -Eulerian polynomials

Theorem (S, Viswanathan 2012)

For positive integer k,

An,k (x) =∑

e∈In,k

xasc e

∑t≥0

(t − 1 + 1

kt

)(kt + 1)nx t =

An,k (x)

(1− x)n+ 1k∑

n≥0

An,k (x)zn

n!=

(1− x

ekz(x−1) − x

) 1k

Theorem (S, Viswanathan 2012)

∑e∈In,k

x asc e =∑π∈Sn

x exc πkn−#cyc π,

where

excπ = |{i | π(i) > i}|

and #cycπ is the number of cycles in the disjoint cyclerepresentation of π.

Combinatorial proof?

Theorem (S, Viswanathan 2012)

∑e∈In,k

x asc e =∑π∈Sn

x exc πkn−#cyc π,

where

excπ = |{i | π(i) > i}|

and #cycπ is the number of cycles in the disjoint cyclerepresentation of π.

Combinatorial proof?

7.

Lecture Hall Partitions

Ln =

{λ ∈ Zn | λ1

1≤ λ2

2≤ · · · ≤ λn

n

}

7.

Lecture Hall Partitions

Ln =

{λ ∈ Zn | λ1

1≤ λ2

2≤ · · · ≤ λn

n

}

s-lecture hall partitions

L(s)n =

{λ ∈ Zn | λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn

}

Theorem (Bousquet-Mélou, Eriksson 1997)

For s = (1,2, . . .n),∑λ∈L(s)

n

q|λ| =1

(1− q)(1− q3) · · · (1− q2n−1),

where |λ| = λ1 + · · ·+ λn.

(What other sequences s give rise to nice generatingfunctions?)

s-lecture hall partitions

L(s)n =

{λ ∈ Zn | λ1

s1≤ λ2

s2≤ · · · ≤ λn

sn

}

Theorem (Bousquet-Mélou, Eriksson 1997)

For s = (1,2, . . .n),∑λ∈L(s)

n

q|λ| =1

(1− q)(1− q3) · · · (1− q2n−1),

where |λ| = λ1 + · · ·+ λn.

(What other sequences s give rise to nice generatingfunctions?)

8.Fundamental Lecture Hall Parallelepiped

s = (3,5)

s = (2,4,6)

Fundamental (half-open) s-lecture hall parallelepiped:

Π(s)n =

{n∑

i=1

ciwi | 0 ≤ ci < 1

},

where wi = [0, . . . ,0, si , si+1, . . . sn].

Theorem (Liu, Stanley 2012)

There is a bijection between I(s)n and Π

(s)n ∩ Zn.

I(3,5)2

→

Π(3,5)2 ∩ Z2

I(2,4,6)3

→

‘Π

(2,4,6)3 ∩ Z3

I(3,5)2

→

Π(3,5)2 ∩ Z2

I(2,4,6)3

→

‘Π

(2,4,6)3 ∩ Z3

9.

Inflated s-Eulerian polyomials

∑λ∈Π

(s)n

xλn

Define the inflated s-Eulerian polyomial by

Q(s)n (x) =

∑λ∈Π

(s)n

xλn .

Theorem (Pensyl,S 2013)

For any sequence s of positive integers,

Q(s)n (x) =

∑e∈I(s)

n

xsnasc e−en

For s = (1,2, . . . ,n), the coefficient sequence of Q(s)n gives an

interesting refinement of the Eulerian numbers:

Coefficient sequence:

1, 1, 2, 4, 4, 4, 4, 2, 1, 1

For s = (1,3, . . . ,2n − 1), the coefficient sequence of Q(s)n (x) is

symmetric (but not the coefficient sequence of A(s)n (x).)

Coefficient sequence:

1, 1, 2, 4, 4, 6, 9, 10, 10, 11, 10, 10, 9, 6, 4, 4, 2, 1, 1

For s = (1,1,2,3,5,8, . . .), the coefficient sequence of Q(s)n is

not symmetric for n ≥ 5.

Coefficient sequence:

1, 1, 2, 2, 4, 4, 4, 4, 4, 2, 1, 1,

10.

Gorenstein cones and self-reciprocal generating functions

Self-reciprocal:

satisfies f (q) = qbf (1/q) for some nonnegative integer b

Examples:

1 + x + 2x2 + 4x3 + 4x4 + 4x5 + 4x6 + 2x7 + x8 + x9

1(1− q)(1− q3)(1− q5)

A pointed rational cone C ⊆ Rn is Gorenstein if there exists apoint c in the interior C0 of C such that C0 ∩ Zn = c + (C ∩ Zn).

Theorem (Special case of a result due to Stanley 1978)

The s-lecture hall cone is Gorenstein if and only if Q(s)n (x) is

self-reciprocal; also, if and only if the following is self reciprocal:

f (s)n (q) =

∑λ∈L(s)

n

q|λ|

Theorem (Bousquet-Mélou, Eriksson 1997; Beck, Braun,Köppe, S, Zafeirakopoulos 2012)

The s-lecture hall cone is Gorenstein if and only if there existsc ∈ Zn satisfying

cjsj−1 = cj−1sj + gcd(sj , sj−1)

for j > 1 with c1 = 1.

Theorem (BBKSZ)

[Beck, Braun, Köppe, S, Zafeirakopoulos 2012]

Let s be a sequence of positive integers defined by

sn = `sn−1 + msn−2, (∗)

with s0 = 0, s1 = 1. Then the s-lecture hall cone in Gorensteinfor all n if and only if m = −1.

Sequences (*) with m = −1 are called `-sequences.

Theorem (BBKSZ)

[Beck, Braun, Köppe, S, Zafeirakopoulos 2012]

Let s be a sequence of positive integers defined by

sn = `sn−1 + msn−2, (∗)

with s0 = 0, s1 = 1. Then the s-lecture hall cone in Gorensteinfor all n if and only if m = −1.

Sequences (*) with m = −1 are called `-sequences.

`-sequences

sn = `sn−1 − sn−2,

with s0 = 1, s1 = 1.

` = 2

1, 2, 3, 4, 5, 6, 7, 8, 9, . . .

` = 3

1, 3, 8, 21, 55, 144, 377, 987, 2584, . . .

Theorem (Bousquet-Mélou, Eriksson 1997)

If s is an `-sequence,∑λ∈L(s)

n

q|λ| =1

(1− q)(1− qs1+s2)(1− qs2+s3) · · · (1− qsn−1+sn ).

Conversely, by the BBKSZ Theorem, for a sequence of the form(*) unless s is an `-sequence, the s-lecture hall partitionscannot, for all n, have a generating function of the form

1(1− qc1)(1− qc2) · · · (1− qcn )

.

Question

What combinatorial family is being represented by thes-inversion sequences when s is an `-sequence?

(When ` = 2, the answer is permutations.)

References

1. s-Lecture hall partitions, self-reciprocal polynomials, andGorenstein cones, M. Beck, B, Braun, M. Köppe, C. D. Savage,and Z. Zafeirakopoulos, submitted.

2. Lecture hall partitions, M. Bousquet-Mélou and K. Eriksson,Ramanujan J., 1(1):101-111, 1997.

3. Lecture hall partitions. II, M. Bousquet-Mélou and K.Eriksson, Ramanujan J., 1(2):165-185, 1997.

4. q-Eulerian polynomials arising from Coxeter groups, F.Brenti, European J. Comb., 15:417.441, September 1994.

5. The Lecture Hall Parallelepiped, F. Liu and R. P. Stanley, July2012, arXiv:1207.6850

6. Rational lecture hall polytopes and inflated Eulerianpolynomials, T. W. Pensyl and C. D. Savage, Ramanujan J., Vol.31 (2013) 97-114.

7. Lecture hall partitions and the wreath products Ck o Sn, T. W.Pensyl and C. D. Savage, Integers, 12B, #A10 (2012/13).

8. Ehrhart series of lecture hall polytopes and Eulerianpolynomials for inversion sequences, C. D. Savage and M. J.Schuster, Journal of Combinatorial Theory, Series A, Vol. 119(2012) 850-870.

9. The 1/k-Eulerian polynomials, C. D. Savage and G.Viswanathan, The Electronic Journal of Combinatorics, Vol. 19(2012) Research Paper P9 , 21 pp. (electronic).

10. The Eulerian polynomials of type D have only real roots, C.D. Savage and M. Visontai, FPSAC 2013, Paris.

11. The s-Eulerian polynomials have only real roots, C. D.Savage and M. Visontai, submitted, arXiv:1208.3831, August2012.

12. Hilbert functions of graded algebras, R. P. Stanley,Advances in Math., 28(1):57.83, 1978.

Thank you!