Gamma-Nonnegativity in Enumerative and GeometricCombinatorics

Christos Athanasiadis

University of Athens

May 21, 2016

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Outline

1 Definitions

2 Gamma-nonnegativity in enumerative combinatorics

3 Gamma-nonnegativity in geometric combinatorics

4 Methods

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Symmetry and unimodality

Definition

A polynomial f (x) ∈ R[x ] is

• symmetric (or palindromic) and• unimodal

if for some n ∈ N,

f (x) = p0 + p1x + p2x2 + · · ·+ pnx

n

with

• pk = pn−k for 0 ≤ k ≤ n and• p0 ≤ p1 ≤ · · · ≤ pbn/2c.

The number n/2 is called the center of symmetry.

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Example: Eulerian polynomial

We let

• Sn be the group of permutations of [n] := {1, 2, . . . , n}

and for w ∈ Sn

• des(w) := # {i ∈ [n − 1] : w(i) > w(i + 1)}• exc(w) := # {i ∈ [n − 1] : w(i) > i}

be the number of descents and excedances of w , respectively. Thepolynomial

An(x) :=∑w∈Sn

xdes(w) =∑w∈Sn

xexc(w)

is the nth Eulerian polynomial.

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Example

An(x) =

1, if n = 1

1 + x , if n = 2

1 + 4x + x2, if n = 3

1 + 11x + 11x2 + x3, if n = 4

1 + 26x + 66x2 + 26x3 + x4, if n = 5

1 + 57x + 302x2 + 302x3 + 57x4 + x5, if n = 6.

Note: The Eulerian polynomial An(x) is well known to be symmetric andunimodal. Is there a simple combinatorial proof?

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Gamma-nonnegativity

Proposition (Branden, 2004, Gal, 2005)

Suppose f (x) ∈ R[x ] has nonnegative coefficients and only real roots andthat it is symmetric, with center of symmetry n/2. Then

f (x) =

bn/2c∑i=0

γi xi (1 + x)n−2i

for some nonnegative real numbers γ0, γ1, . . . , γbn/2c.

Definition

The polynomial f (x) is called γ-nonnegative if there exist nonnegative realnumbers γ0, γ1, . . . , γbn/2c as above, for some n ∈ N.

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Example

An(x) =

1, if n = 1

1 + x , if n = 2

(1 + x)2 + 2x , if n = 3

(1 + x)3 + 8x(1 + x), if n = 4

(1 + x)4 + 22x(1 + x)2 + 16x2, if n = 5

(1 + x)5 + 52x(1 + x)3 + 186x2(1 + x), if n = 6.

Note: Every γ-nonnegative polynomial (even if it has nonreal roots) issymmetric and unimodal.

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An index i ∈ [n] is called a double descent of a permutation w ∈ Sn if

w(i − 1) > w(i) > w(i + 1),

where w(0) = w(n + 1) = n + 1.

Theorem (Foata–Schutzenberger, 1970)

We have

An(x) =

b(n−1)/2c∑i=0

γn,i xi (1 + x)n−1−2i ,

where γn,i is the number of w ∈ Sn which have no double descent anddes(w) = i . In particular, An(x) is symmetric and unimodal.

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Recall that a permutation w ∈ Sn is said to be up-down if

w(1) < w(2) > w(3) < · · · .

Corollary

We have

An(−1) =

{0, if n is even,

(−1)(n−1)/2 γn,(n−1)/2, if n is odd,

where γn,(n−1)/2 is the number of up-down permutations in Sn.

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Recently, gamma-nonnegativity attracted attention after the work of

• Branden (2004, 2008) on P-Eulerian polynomials,• Gal (2005) on flag triangulations of spheres.

A book exposition can be found in:

• T.Kyle Petersen, Eulerian Numbers, Birkhauser, 2015.

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P-Eulerian polynomials

We let

• P be a poset with n elements,• ω : P → [n] be an order preserving bijection.

Definition (Stanley, 1972)

The P-Eulerian polynomial is defined as

WP(x) =∑

w∈L(P,ω)

xdes(w),

where L(P, ω) consists of all permutations (a1, a2, . . . , an) ∈ Sn with theproperty

ω−1(ai ) <P ω−1(aj) ⇒ i < j .

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Example

For

(P, ω) = ss

ss

1 2

3 4

���

we have

L(P, ω) = {(1, 2, 3, 4), (1, 2, 4, 3), (2, 1, 3, 4), (2, 1, 4, 3), (1, 3, 2, 4)}

and

WP(x) = 1 + 3x + x2.

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Example

For an n-element antichain P (no two elements are comparable)

(P, ω) = s s s s1 2 3 4

we have

L(P, ω) = Sn

and hence

WP(x) = An(x).

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Note: The polynomial WP(x):

• plays a role in Stanley’s theory of P-partitions,• does not depend on ω,• is symmetric, provided P is graded,• can have non-real roots, as shown by Branden and Stembridge.

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Theorem (Reiner–Welker, 2005)

The polynomial WP(x) is unimodal for every graded poset P.

Their proof uses deep results from geometric combinatorics. Branden gavetwo elementary proofs of the following:

Theorem (Branden, 2004, 2008)

The polynomial WP(x) is γ-nonnegative for every graded poset P.

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Derangement polynomials

We let Dn be the set of derangements in Sn. The polynomial

dn(x) :=∑w∈Dn

xexc(w)

is the nth derangement polynomial.

Example

dn(x) =

0, if n = 1

x , if n = 2

x + x2, if n = 3

x + 7x2 + x3, if n = 4

x + 21x2 + 21x3 + x4, if n = 5

x + 51x2 + 161x3 + 51x4 + x5, if n = 6,

x + 113x2 + 813x3 + 813x4 + 113x5 + x6, if n = 7.

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Note: The unimodality of dn(x) follows from deep results of Stanley onlocal h-polynomials of triangulations of simplices. Other proofs of uni-modality were given by:

• Brenti (1990),• Stembridge (1992),• Zhang (1995).

Note:

dn(x) =

0, if n = 1

x , if n = 2

x(1 + x), if n = 3

x(1 + x)2 + 5x2, if n = 4

x(1 + x)3 + 18x2(1 + x), if n = 5

x(1 + x)4 + 47x2(1 + x)2 + 61x3, if n = 6

x(1 + x)5 + 108x2(1 + x)3 + 479x3(1 + x), if n = 7.

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A descending run of a permutation w ∈ Sn is a maximal string of indices{a, a + 1, . . . , b} such that w(a) > w(a + 1) > · · · > w(b). An index i ∈[n − 1] is a double excedance of w if w(i) > i > w−1(i).

Theorem

We have

dn(x) =

bn/2c∑i=0

ξn,i xi (1 + x)n−2i ,

where ξn,i equals the number of:

• permutations w ∈ Sn with i runs and no run of size one,• derangements w ∈ Dn with i excedances and no double excedance.

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This statement, along with several q-analogues and generalizations, wasdiscovered independently (using different methods) by:

• A–Savvidou (2012),• Gessel–Shareshian–Wachs (2010),• Linusson–Shareshian–Wachs (2012),• Shin–Zeng (2012),• Sun–Wang (2014).

For instance:

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We denote by c(w) the number of cycles of w ∈ Sn.

Theorem (Shin–Zeng, 2012)

We have

∑w∈Dn

qc(w)xexc(w) =

bn/2c∑i=0

ξn,i (q) x i (1 + x)n−2i ,

where

ξn,i (q) =∑

w∈Dn(i)

qc(w)

and Dn(i) consists of all elements of Dn with exactly i cyclic valleys andno double cyclic descent.

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Recall that

maj(w) =∑

i∈Des(w)

i

is the major index of w ∈ Sn.

Theorem (Gessel–Shareshian–Wachs, 2010)

We have

∑w∈Dn

pmaj(w)−exc(w)qdes(w)xexc(w) =

bn/2c∑i=0

ξn,i (p, q) x i (1 + x)n−2i

for some polynomials ξn,i (p, q) in p, q with nonnegative coefficients.

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Corollary

We have

dn(−1) =

{0, if n is odd,

(−1)n/2 ξn,n/2, if n is even,

where ξn,n/2 is the number of up-down permutations in Sn.

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Involutions

We let In be the set of permutations w ∈ Sn with w = w−1 and let

In(x) :=∑w∈In

xdes(w).

Example

In(x) =

1, if n = 1

1 + x , if n = 2

1 + 2x + x2, if n = 3

1 + 4x + 4x2 + x3, if n = 4

1 + 6x + 12x2 + 6x3 + x4, if n = 5

1 + 9x + 28x2 + 28x3 + 9x4 + x5, if n = 6,

1 + 12x + 57x2 + 92x3 + 57x4 + 12x5 + x6, if n = 7.

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Note: The polynomial In(x) was first considered by Brenti.

Theorem (Guo–Zeng, 2006)

The polynomial In(x) is symmetric and unimodal for every n.

The proof uses generating functions and recursions.

Conjecture (Guo–Zeng, 2006)

The polynomial In(x) is γ-nonnegative for every n.

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Example

In(x) =

1, if n = 1

1 + x , if n = 2

(1 + x)2, if n = 3

(1 + x)3 + x(1 + x), if n = 4

(1 + x)4 + 2x(1 + x)2 + 2x2, if n = 5

(1 + x)5 + 4x(1 + x)3 + 6x2(1 + x), if n = 6

(1 + x)6 + 6x(1 + x)4 + 18x2(1 + x)2, if n = 7.

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More examples

There is an endless list of generalizations and similar results, including:

• q-analogues,• Coxeter group analogues,• analogues for colored permutations,• results for other interesting classes of permutations.

More examples to follow...

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Face enumeration of simplicial complexes

We let

• ∆ be a simplicial complex of dimension n − 1,• fi (∆) be the number of i-dimensional faces.

Definition

The h-polynomial of ∆ is defined as

h(∆, x) =n∑

i=0

fi−1(∆) x i (1− x)n−i =n∑

i=0

hi (∆) x i .

The sequence h(∆) = (h0(∆), h1(∆), . . . , hn(∆)) is the h-vector of ∆.

Note: h(∆, 1) = fn−1(∆).

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Example

For the 2-dimensional complex

∆ =

s ss

ss

ss

s����������

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@@@

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we have f0(∆) = 8, f1(∆) = 15 and f2(∆) = 8 and hence

h(∆, x) = (1− x)3 + 8x(1− x)2 + 15x2(1− x) + 8x3

= 1 + 5x + 2x2.

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Theorem (Klee, Reisner, Stanley)

The polynomial h(∆, x):

• has nonnegative coefficients if ∆ triangulates a ball or a sphere,

• is symmetric if ∆ triangulates a sphere,

• is unimodal if ∆ is the boundary complex of a simplicial polytope.

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Example

We let

• V be an n-element set,• 2V be the simplex on the vertex set V ,• Γ be the first barycentric subdivision of the boundary complex of 2V .

Then h(Γ, x) = An(x). For n = 3

Γ = s ss

ss

s�����@

@@@@

h(∆, x) = (1− x)2 + 6x(1− x) + 6x2 = 1 + 4x + x2.

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Flag complexes and Gal’s conjecture

Definition

A simplicial complex ∆ is called flag if it contains every simplex whose1-skeleton is a subcomplex of ∆.

ss

s

s�����

����������

HHHHH

AAAAAAAAAA

not flag

s ss

ss

ss

s����������

�����

@@@

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����

flag

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Example

For a 1-dimensional sphere ∆ with m vertices we have

h(∆, x) = 1 + (m − 2)x + x2.

Note that h(∆, x) is γ-nonnegative ⇔ m ≥ 4 ⇔ ∆ is flag.

Conjecture (Gal, 2005)

The polynomial h(∆, x) is γ-nonnegative for every flag triangulation ∆ ofthe sphere.

Note: This extends a conjecture of Charney–Davis (1995).

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Note: Let us write

h(∆, x) =

bn/2c∑i=0

γi (∆) x i (1 + x)n−2i .

Then Gal’s conjecture implies that h2(∆) is bounded below by the co-efficient of x2 in

(1 + x)n + γ1(∆)x(1 + x)n−2,

which means the following:

Conjecture

Among all flag triangulations of the (n − 1)-dimensional sphere with givennumber m of vertices, the (n− 2)-fold double suspension over the complexof an (m − 2n + 4)-gon has the smallest possible number of edges.

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Note: By a result of Karu (2006), Gal’s conjecture holds for barycentricsubdivisions of regular CW-spheres.

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The local h-polynomial

We let

• V be an n-element set,• Γ be a triangulation of the simplex 2V on the vertex set V .

Definition (Stanley, 1992)

The local h-polynomial of Γ (with respect to V ) is defined as

`V (Γ, x) =∑F⊆V

(−1)n−|F | h(ΓF , x),

where ΓF is the restriction of Γ to the face F of the simplex 2V .

Note: This polynomial plays a major role in Stanley’s theory of subdivisi-ons of simplicial (and more general) complexes.

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Example

For the 2-dimensional triangulation

Γ =

ss

sss

ss

s����������

�����

@@@

��������AAAAAAAAAA�

����

we have

`V (Γ, x) = (1 + 5x + 2x2) − (1 + 2x) − (1 + x) − 1

+ 1 + 1 + 1 − 1 = 2x + 2x2.

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Theorem (Stanley, 1992)

The polynomial `V (Γ, x)

• is symmetric,• has nonnegative coefficients,• is unimodal for every regular triangulation Γ of 2V .

Conjecture (A, 2012)

The polynomial `V (Γ, x) is γ-nonnegative, if Γ is a flag triangulation of 2V .

Note: this is stronger than Gal’s conjecture. There is considerable eviden-ce for both conjectures.

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We denote by Σn the boundary complex of the n-dimensional cross-poly-tope.

Theorem (A, 2012)

Every flag triangulation ∆ of the (n − 1)-dimensional sphere is a flag, ve-rtex-induced homology subdivision Γ of Σn. Moreover,

h(∆, x) =∑F∈Σn

`F (ΓF , x) (1 + x)n−|F |,

hence the γ-nonnegativity of h(∆, x) is implied by that of the `F (ΓF , x).

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Corollary

For every flag triangulation ∆ of the (n − 1)-dimensional sphere,

h(∆, x) ≥ (1 + x)n

holds coefficientwise.

Note: This holds, more generally, for doubly Cohen–Macaulay flag comple-xes of dimension n − 1.

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Example

For the barycentric subdivision Γ of the simplex 2V on the vertex set V

{1}

{3}{2} {2, 3}

{1, 2} {1, 3}

{1, 2, 3}

Stanley showed that

`V (Γ, x) =n∑

k=0

(−1)n−k(n

k

)Ak(x) =

∑w∈Dn

xexc(w) = dn(x).

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More barycentric subdivisions

Consider the barycentric subdivision K of the cubical barycentric subdivi-sion of the simplex 2V .

Note: The sum of the coefficients of `V (K , x) is equal to the number of

• even derangements in Bn,• derangements in Dn,

where Bn is the group of signed permutations of [n] and Dn is the sub-group of even signed permutations.

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Example

`V (K , x) =

0, if n = 1

3x , if n = 2

7x + 7x2, if n = 3

15x + 87x2 + 15x3, if n = 4

31x + 551x2 + 551x3 + 31x4, if n = 5

63x + 2803x2 + 8243x3 + 2803x4 + 63x5, if n = 6.

Conjecture

The polynomial `V (K , x) has only real roots.

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For w = (w1,w2, . . . ,wn) ∈ Bn we let

• excA(w) := # {i ∈ [n − 1] : w(i) > i},• neg(w) := # {i ∈ [n] : w(i) < 0}.

Definition (Bagno–Garber, 2006)

The flag-excedance number of w ∈ Bn is defined as

fex(w) = 2 · excA(w) + neg(w).

Example: For

• w = (3,−5, 1, 4,−2)

we have excA(w) = 1 and neg(w) = 2, so fex(w) = 4.

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Theorem (A, 2014)

We have

`V (K , x) =∑w

x fex(w)/2

=

bn/2c∑i=0

ξ+n,i x

i (1 + x)n−2i ,

where the first sum runs over all derangements w ∈ Dn and ξ+n,i is the

number of elements of Bn with i decreasing runs, none of size one, andpositive last coordinate.

Problem: Find a simple combinatorial proof of the second expression.

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Consider the second barycentric subdivision Γ2 of the simplex 2V .

Note: The sum of the coefficients of `V (Γ2, x) equals the number of pairs(u, v) ∈ Sn ×Sn of permutations with no common fixed point.

Problem: Find a combinatorial interpretation for:

• the coefficients of `V (Γ2, x),• the coefficients in the expansion

`V (Γ2, x) =∑

γi xi (1 + x)n−2i .

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Problem: Study the barycentric subdivision of more general polyhedralsubdivisions of the simplex.

For a generalizations and other results of this type see

• C.A. Athanasiadis, A survey of subdivisions and local h-vectors, in”The Mathematical Legacy of Richard P. Stanley”, AMS, 20xx.

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Methods

Methods to prove γ-nonnegativity include:

• valley hopping (Foata–Schutzenberger–Strehl)• combinatorial expansions (Branden, Shin–Zeng, Stembridge)• enriched P-partitions (Stembridge, Petersen)• poset homology, shellability (Linusson–Shareshian–Wachs)• symmetric functions (Shareshian–Wachs),• geometric methods (A–Savvidou).

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