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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!