Box Polynomials of Lattice Simplices

IN

DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2018,

Box Polynomials of Lattice Simplices

NILS GUSTAFSSON

KTH ROYAL INSTITUTE OF TECHNOLOGY

Box Polynomials of Lattice Simplices

NILS GUSTAFSSON

Degree Projects in Mathematics (30 ECTS credits) Degree Programme in Mathematics (120 credits) KTH Royal Institute of Technology year 2018 Supervisor at KTH: Liam Solus

Examiner at KTH: Svante Linusson

TRITA-SCI-GRU 2018:160 MAT-E 2018:24

Royal Institute of Technology School of Engineering Sciences KTH SCI

SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

The box polynomial of a lattice simplex is a variant of the more well-known h^∗-polynomial, where the open fundamental parallelepiped is considered instead of the half-open. Box polynomials are connected to h^∗- polynomials by a theorem of Betke and McMullen from 1985. This theorem can be used to prove certain properties of h^∗-polynomials, such as unimodality and symmetry.

In this thesis, we investigate box polynomials of a certain family of simplices, called s-lecture hall simplices.

The h^∗-polynomials of these simplices are a generalization of Eulerian polynomials, and were proven to be real-rooted by Savage and Visontai in 2015. We use a modified version of their proof to prove that the box polynomials are also real-rooted, and show that they are a generalization of derangement polynomials. We then use these results to partially answer a conjecture by Br¨and´en and Leander regarding unimodality of h^∗-polynomials of s-lecture hall order polytopes.

Sammanfattning

Boxpolynomet av ett gittersimplex ¨ar en variant av det mer k¨anda h^∗-polynomet, d¨ar den ¨oppna funda- mentala parallelepipeden anv¨ands ist¨allet f¨or den halv¨oppna. Boxpolynom ¨ar kopplade till h^∗-polynom tack vare en sats av Betke och McMullen fr˚an 1985. Denna sats kan anv¨andas f¨or att bevisa vissa egenskaper hos h^∗-polynom, som t.ex. unimodalitet och symmetri.

I denna uppsats unders¨oker vi boxpolynomen hos en s¨arskild familj av simplex, de s˚a kallade s-h¨orssalssim- plexen. F¨or dessa simplex ¨ar h^∗-polynomen en generalisering av de Eulerska polynomen, och visades ha endast reella r¨otter av Savage och Visontai 2015. Vi anv¨ander en modifierad version av deras bevis f¨or att bevisa att ¨aven boxpolynomen bara har reella r¨otter, och att de ¨ar en generalisering av derangemangpoly- nom. Vi anv¨ander sedan dessa resultat f¨or att delvis besvara en f¨ormodan av Br¨and´en och Leander ang˚aende unimodaliteten hos h^∗-polynomen av s-h¨orsalsordningspolytoper.

Contents

1 Introduction 4

2 Ehrhart theory 6

2.1 Polytopes . . . 6

2.2 Ehrhart’s theorem . . . 8

2.3 Distributional properties of h^∗-polynomials . . . 12

2.4 Triangulations and Box Polynomials . . . 13

3 s-Eulerian Polynomials 16 3.1 Lehmer codes . . . 16

3.2 s-inversion sequences . . . 17

3.3 s-lecture hall simplices . . . 19

4 s-Derangement Polynomials 22 4.1 The box polynomial of an s-lecture hall polytope . . . 22

4.2 Derangements and the sequence s = (n, n − 1, n − 2, ..., 2) . . . 23

4.3 Colored permutations . . . 25

4.4 Real-rootedness . . . 30

4.5 Faces of s-lecture hall simplices . . . 34

5 s-Lecture Hall Order Polytopes 37 5.1 Order polytopes . . . 37

5.2 s-lecture hall order polytopes . . . 38

Chapter 1

Introduction

Polytopes are fundamental geometric objects that have been studied since ancient times. Informally, a poly- tope is an object with “flat sides” and “sharp edges”. Examples include cubes, triangles and intervals. A circle is not a polytope. Polytopes are useful in a wide variety of situations. For example, they can be used to approximate things that are not polytopes, like in computer graphics, or to describe the set of possible solutions to an optimization problem, like in linear programming.

Most people would agree that a polytope is an object of continuous nature, as opposed to say a graph or a list of positive integers. However, polytopes can also be useful in more discrete settings in combinatorics and number theory. One such link between polytopes (in the case of convex lattice polytopes) and combinatorics is Ehrhart theory. Ehrhart theory was developed by Eug`ene Ehrhart in the 1960:s, and concerns the so called discrete volume of polytopes. The discrete volume of a polytope is the number of points with integer coordinates contained in it. When the polytope is scaled up by integer factors and the discrete volume is computed for each of scaled up versions, a sequence of numbers is obtained. Ehrhart showed in [14] that this sequence can be described by a polynomial, called the Ehrhart polynomial. Not only that, but this polynomial contains the continuous volume as one of its coefficients, as well as a lot of other information.

The Ehrhart polynomial can also be written in terms of another polynomial, called the h^∗-polynomial. In the case of a simplex, the h^∗-polynomial has a simple geometric interpretation, which makes it relatively easy to compute. For other polytopes, it is harder to interpret what the coefficients of the h^∗-polynomial mean, but a theorem due to Stanley [26] says that the coefficients are always, at least, nonnegative integers.

Sometimes it is discovered that a sequence of integers from combinatorics is the h^∗-polynomial of some polytope. When that happens, Ehrhart theory acts as a kind of bridge between the continuous world of polytopes and the discrete world of combinatorics, which can be of great benefit for both sides. One such example are the Eulerian numbers. The Eulerian number An,k is the number of permutations of [n] with k descents, i.e. k places where π_i> π_i+1. If we let the sequence {A_n,0, A_n,1, · · · A_n,n−1} be the coefficients of a polynomial, then it is also the h^∗-polynomial of a simplex called the lecture hall simplex. This example can be generalized into a rather broad family of polynomials called s-Eulerian polynomials, which are the h^∗-polynomials of s-lecture hall simplices.

When we have a sequence of positive integers that describe a distribution such as “the number of permu- tations with a certain number of descents”, it is often interesting to know whether the sequence is unimodal or not. Unimodal means that the sequence only has one “peak” (or local maximum). It turns out that unimodality often follows if the polynomial corresponding to the sequence is real-rooted. In [23], Savage and Visontai used tools from algebra to show that the s-Eulerian polynomials are real-rooted, and therefore unimodal.

Figure 1.1: A dodecahedron is a polytope.

In this thesis, we will investigate a variant of the h^∗-polynomials of lattice simplices, called box polynomials.

Like h^∗-polynomials, we will see in this thesis that box polynomials can also correspond to interesting com- binatorial sequences. Due to a theorem of Betke and McMullen [8], the h^∗-polynomial of a polytope can be written as a kind of weighted sum of these box polynomials. Sometimes, we can then use unimodality of the box polynomials to recover unimodality of the h^∗-polynomial.

In Chapter 2, Ehrhart’s theory will be introduced in more detail, as well as some basics on polytopes and unimodality. Most of the Ehrhart theory is based on [7], and we will use the same notation. In Chapter 3, we will introduce s-inversion sequences, s-lecture hall polytopes, and s-Eulerian polynomials. Most definitions and results are collected from various articles, such as [23], [22] and [18]. Our definitions and notation may differ slightly as compared to them. For example, we use a different definition of descents compared to the one in [22]. In Chapter 4, we study the properties of box polynomials of s-lecture hall simplices. First, the theory from Chapter 3 is used to describe them in terms of s-inversion sequences. Then, we turn our attention to permutations and colored permutations, where the box polynomials are shown to be a well- known family of polynomials called derangement polynomials. Then, we use a modified version of the proof in [23] to show that they are real-rooted. Finally, we show that the faces of s-lecture hall simplices also have real-rooted box polynomials. In Chapter 5, we apply the real-rootedness results from Chapter 4 combined with the theorem of Betke-McMullen to s-lecture hall order polytopes, a very broad family of polytopes introduced in [12]. We show that the h^∗-polynomials for some of these polytopes are unimodal.

Chapter 2

Ehrhart theory

Ehrhart theory deals with the problem of counting lattice points in convex polytopes. It provides a kind of bridge between geometry and combinatorics. This chapter contains some Ehrhart theory that will be used throughout the article. For a comprehensive discussion on the topic we refer the reader to [7].

2.1 Polytopes

A convex polytope P in Rⁿ is the convex hull of finitely many points:

P = {λ1v1+ ... + λmvm∈ Rⁿ : λi ≥ 0 and λ1+ ... + λm= 1}.

The points v₁, ..., v_mare called the vertices of P. If the vertices have integer coordinates, then we say that P is a lattice polytope (or integral polytope). The dimension of P is the dimension of the affine space

span(P) = {x + λ(y − x) : x, y ∈ P, λ ∈ R}.

If P has dimension d, then P is called a d-polytope. The definition above of a polytope as the convex hull of finitely many points is called the vertex description of P. Every polytope also has an equivalent description as the bounded intersection of finitely many half-spaces and hyperplanes. This is called the hyperplane description. These descriptions are useful in different situations, and we will often use both of them. The interior of a polytope P, denoted P^◦, is the set of points of P such that all points within distance  from them also belong to P, for some  > 0. The interior can be obtained by taking the hyperplane description of P and turning all inequalities strict.

A face of a convex polytope is a set on the form P ∩ H, where H is a hyperplane such that P lies entirely on one side of it. The (d − 1)-dimensional faces are called facets, and the 2-dimensional faces are called edges. A d-simplex is a d-polytope with exactly d + 1 vertices. This is the smallest number of vertices a d-polytope can have, because otherwise it can’t have dimension d. A 1-simplex is an interval, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. Simplices are often easier to work with than general polytopes.

Therefore, it can be useful to decompose a polytope into simplices.

Definition 2.1.1. Let S ⊆ Rⁿ. A triangulation of S is a finite set of simplices T such that 1. S =S

∆∈T∆,

2. If ∆ ∈ T and ∆⁰ is a face of ∆, then ∆⁰ ∈ T , and 3. If ∆ ∈ T and ∆⁰ ∈ T , then ∆ ∩ ∆⁰ ∈ T .

If the vertices of the simplices in T are the same as the vertices of a polytope P, then we say that P is tri- angulated using no new vertices. All convex polytopes can be triangulated using no new vertices, because

Figure 2.1: A convex 2-polytope along with its second and third dilates. The polytope is the convex hull of (1, 1), (0, −1) and (−1, 0).

a triangulation can be constructed in the following way: Take the convex d-polytope P = conv(v1, ..., vn), and lift it to (d + 1)-dimensional space by adding an extra coordinate to every vertex. Call this new polytope P⁰. This lifting can be done in such a way that the vertices of P⁰ are in general position. Now, take the convex hull of P⁰. Since the vertices are in general position, the facets of the convex hull will be simplices. If we take the lower hull of this (the facets such that all points “below” them in the xd+1-direction are outside of the convex hull), and move it back to R^d by deleting the last coordinate, we get a triangulation of P. A triangulation obtained in this way is called a regular triangulation, and they will be of special interest later.

One main problem of Ehrhart theory is to count the lattice points in a polytope when it is scaled up by different factors. For a convex polytope P and a positive integer t, the t:th dilate of P is defined as:

tP := {tx ∈ Rⁿ : x ∈ P}.

For positive integers t, define the lattice-point enumerator of P to be the number of lattice points in the t:th dilate of P:

L_P(t) = #(tP ∩ Zⁿ).

The generating function containing L_P(t) as coefficients is called the Ehrhart series of P:

EhrP(z) = 1 +X

t≥1

LP(t)z^t.

Example 2.1.2. One of the most important examples of a polytope is the unit cube ^d:

^d:= [0, 1]^d= conv(x1, ..., xd) ∈ R^d : all xi are 0 or 1 .

^d can be written as the intersection of 2d half-spaces.

^d =

d

\

i=1

{xi≥ 0} ∩ {xi≤ 1}.

Here {x_i≥ 0} is short for {(x1, ..., x_d) ∈ R^d : x_i ≥ 0}. This is the hyperplane description of d. The t:th dilate of d is

td= [0, t]^d, which implies that

L_d(t) = (t + 1)^d, and

Ehr_d(z) =X

t≥0

(t + 1)^dz^t.

Figure 2.2: The unit cube 2 along with its second, third, and fourth dilates.

2.2 Ehrhart’s theorem

In this section, some motivation behind the definition of LP(t) and EhrP(z) will be provided. Both func- tions turn out to be very useful, as they contain a lot of information about the polytope. For example, Ehrhart’s theorem says that LP(t) is a polynomial:

Theorem 2.2.1 ([14]). If P is a convex lattice polytope of dimension d, then L_P(t) is a polynomial in t of degree d.

Due to Theorem 2.2.1, L_P(t) is called the Ehrhart polynomial (not to be confused with the Ehrhart series Ehr_P(z)). Ehrhart’s theorem implies that L_P(t) can be written as

L_P(t) = cdt^d+ cd−1t^d−1+ ... + c1t + c0,

for some rational coefficients c0, c1, · · · , cd. Now we could try to make sense of these coefficients. The first one, cd, actually turns out to be a very familiar number: the volume of P. To see why, consider the following quantity:

t→∞lim L_P(t)

t^d = lim

t→∞

#tP ∩ Zⁿ t^d .

The numerator on the right-hand-side is the number of lattice points inside P when P is scaled up by a factor of t. Each such lattice point contributes t^−d to the expression on the right. So it is a bit like we covered P with hypercubes of side length t⁻¹ and added up their volumes. For large t, this should give a good approximation of the P:s volume (in fact, the volume can be defined as the limit above). Finally we note that the expression on the left is cd, so cd really is the volume of P.

This means that once we have found L_P(t), we also know the volume of P. However, there is a lot more information in L_P(t) and Ehr_P(z). To find some of it, we will need some definitions that naturally turn up in the proof of Theorem 2.2.1. It turns out that it suffices to prove Theorem 2.2.1 in the case when P is a simplex. To see this, note that if P is not a simplex, then it can be triangulated using no new vertices, and we can express L_P(t) with the inclusion-exclusion formula

L_P(t) = X

∆∈T

(−1)^d−dim(∆)L∆(t).

Here T is a triangulation using no new vertices. So if Theorem 2.2.1 is true for simplices, then L_P(t) for a general lattice polytope is a sum of polynomials of degree d, and is therefore a polynomial of degree at most d. If the degree was smaller than d, then the volume of P would be 0 using the argument above. However, this would imply that the dimension of P is smaller than d. So the degree of LP(t) must be exactly d.

Figure 2.3: The cone over the interval P = [−1, 2]. Note that the intersection with the hyperplane {y = 1}

is the polytope itself.

To prove Theorem 2.2.1 for simplices, we first need some definitions. For a set S ⊆ Rⁿ, the integer-point transform σ_S(z) is defined as

σ_S(z) := X

m∈S∩Zⁿ

z^m,

where z^m := z₁^m1z₂^m2...z_n^mn. So it is a kind of generating function that lists the lattice points of S. For a convex lattice polytope P with vertices v1, ..., vk, define the cone over P as

cone(P) = {λ₁w₁+ ... + λ_kw_k : all λ_i≥ 0},

where wi:= (vi, 1). Geometrically, we added an extra dimension, lifted P into the hyperplane {xn+1= 1}, made a cone with the origin as the “tip” and made it go through the lifted version of P.

Note that cone(P) ∩ {x_n+1= 1} is just P itself in disguise. This is because if w = λ1w1+ ... + λkwk ∈ cone(P),

then the last coordinate of w is λ₁+ · · · + λ_k. So cone(P) ∩ {x_n+1= 1} is the collection of points of cone(P) where λ₁+ ... + λ_k = 1. This is the convex hull of w₁, ..., w_k, also known as P lifted to {x_n+1 = 1}.

Generalizing this argument a little bit gives the following:

Proposition 2.2.2. For a convex lattice polytope P and a positive integer t, cone(P) ∩ {xn+1= t} = tP.

If we take the integer-point transform of the cone, then interesting things start to happen:

Proposition 2.2.3. For a convex lattice polytope P,

σ_cone(P)(1, 1, ..., 1, z) = Ehr_P(z).

Proof. Recall that the integer-point transform listed all the lattice points of a set S. Take one such lattice point w ∈ cone(P) and see how it contributes to σcone(P)(1, 1, ..., 1, z). The monomial associated to w is z₁^w1...z_n+1^wn+1. When evaluated at (1, 1, ..., 1, z), it becomes z^wn+1, so it contributes 1 to the coefficient in front of z^wn+1 in the series σcone(P)(1, 1, ..., 1, z). However, thanks to Proposition 2.2.2, we know that w corresponds to a point in w_n+1P. So σcone(P)(1, 1, ..., 1, z) enumerates points in tP. In other words,

σ_cone(P)(1, 1, ..., 1, z) = 1 +X

t≥1

L_P(t)z^t= Ehr_P(z).

Now we will express σcone(P)(1, 1, ..., 1, z) in a different way, by tiling it with parallelepipeds.

Definition 2.2.4. For a d-dimensional lattice simplex P, the fundamental parallelepiped Π is defined as

Π := {λ1w1+ ... + λd+1wd+1 : 0 ≤ λi< 1 for all i = 1, 2, ..., d + 1}.

Note that the fundamental parallelepiped is half-open. This is important because it allows us to tile the cone using translated copies of it.

Proposition 2.2.5. Let P be a d-dimensional lattice simplex. For some m ∈ Z^d+1≥0 , let Π_m:= m₁w₁+ ... + m_d+1w_d+1+ Π

be Π translated by some non-negative integer linear combination of w1, ..., wd+1. Then it holds that cone(P) = [

m∈Z^d+1≥0

Πm.

Moreover, the sets Πm are disjoint.

Proof. Let w ∈ cone(P). If we could prove that w belongs to exactly one of the translated parallelepipeds, then the proof would be done. Here, the fact that P is a simplex is crucial for the uniqueness part, because it implies that w₁, ..., w_d+1 are linearly independent. Therefore, w can be uniquely written as

w = λ1w1+ ... + λd+1wd+1.

If we let m = (bλ1c, ..., bλd+1c), then w ∈ Πm, and there is no other such parallelepiped.

Corollary 2.2.6. For a d-dimensional lattice simplex P, σ_cone(P)(z) = X

m∈Z^d+1≥0

σΠ_m(z).

Now we are ready to prove the following:

Proposition 2.2.7. For a d-dimensional lattice simplex P, σ_cone(P)(z) = σ_Π(z)

(1 − z^w1)...(1 − z^wd+1). Proof. First, note that it follows from the definition of σ and Πm that

σΠ_m(z) = z^mσΠ(z).

Combining this with Corollary 2.2.6 yields

σcone(P)(z) = σΠ(z) X

m∈Z^d+1_≥0

z^m= σΠ(z)



 X

i≥0

z^iw1



...



 X

i≥0

z^iwd+1



= σ_Π(z)

(1 − z^w1)...(1 − z^wd+1)

In the last step we used the fact that geometric series satisfiesP

i≥0zⁱ =_1−z¹ . This result can in turn be combined with Proposition 2.2.3:

Proposition 2.2.8. For a d-dimensional lattice simplex P,

Ehr_P(z) = h^∗_dz^d+ h^∗_d−1z^d−1+ ... + h^∗₀ (1 − z)^d+1

where h^∗_k is the number of integer points in Π whose last coordinate is k.

Proof. In Proposition 2.2.7, evaluate at (1, 1, 1, ..., 1, z), to recover σ_cone(P)(1, 1, ..., 1, z) = σΠ(1, 1, ..., 1, z)

(1 − z)^d+1 .

In the proof of Proposition 2.2.3, we noted that evaluating the integer-point transform in (1, 1, ..., 1, z) gives a series that counts the number of integer points with different last coordinates, which is exactly what the h^∗-coefficients do. The reason why the series only goes from 0 to d is that there are no integer points in Π whose last coordinate is outside of this interval. Also, by Proposition 2.2.3, we have that the series above is actually the Ehrhart series. Therefore,

Ehr_P(z) = h^∗_dz^d+ h^∗_d−1z^d−1+ ... + h^∗₀ (1 − z)^d+1 .

The polynomial h^∗_dz^d+ h^∗_d−1z^d−1+ ... + h^∗₀ is called the h^∗-polynomial of the simplex P, and it will become very important later. Now we have everything we need to prove Ehrhart’s theorem:

Proof of Theorem 2.2.1. Rewrite the Ehrhart series using the h^∗-polynomial to see that

Ehr_P(z) = h^∗_dz^d+ h^∗_d−1z^d−1+ ... + h^∗₀ (1 − z)^d+1 =



 X

t≥0

d + t d

 z^t





d

X

i=0

h^∗_izⁱ

!

=X

t≥0 d

X

i=0

d + t − i d

 h^∗_i

! z^t.

By the definition of the Ehrhart series, the expression within the parentheses in the last expression is L_P(t).

Since the expression is a polynomial in t of degree at most d, the proof is almost done. The only thing remaining is to show that the degree is exactly d. This follows from the fact that the volume of P is positive.

To conclude this section, a few more theorems (that will not be proved) and definitions will be presented.

Proofs for each of these results can be found in [14]. These results and definitions will play a key role in the main results of this thesis.

From a combinatorial perspective, one very exciting thing about the h^∗-polynomial of a lattice simplex is that all of its coefficients are nonnegative integers. In fact, the h^∗-polynomial can be defined for general lattice polytopes and not just simplices. This is illustrated by Stanley’s non-negativity theorem:

Theorem 2.2.9 ([26]). For a lattice polytope P,

Ehr_P(z) = h^∗_dz^d+ ... + h^∗₀ (1 − z)^d+1 , where h^∗₀, ..., h^∗_d are non-negative integers.

Now that we know that LP(t) is a polynomial, we can evaluate it at other points than positive integers, and try to interpret what that means. For negative integers, such an interpretation is given by the following reciprocity theorem:

Theorem 2.2.10. If P is a convex lattice polytope with dimension d, then L_P(−t) = (−1)^dL_P^◦(t).

A special kind of polytope that will be useful in Chapter 5 are the reflexive polytopes. Informally, a reflexive polytope is a convex polytope whose vertices have integer coordinates, and whose hyperplane description also only contains integers. More precisely, a lattice d-polytope P is reflexive if it contains the origin in its interior, and can be written as

P = {x ∈ Rⁿ: Ax ≤ 1},

for some matrix A only containing integers. In this thesis, we will say that a polytope is reflexive when it really is reflexive up to translation according to the above definition. This means that it can be translated into a reflexive polytope. The reason for this is that most properties of interest to us, like h^∗- polynomials and regularity of triangulations, are unaffected by translation. Polytopes can be shown to be reflexive by looking at their h^∗-polynomials, using a theorem of Hibi.

Theorem 2.2.11 (Theorem 2.1 in [16]). Let P be a lattice d-polytope. Then P is reflexive if and only if the h^∗-polynomial satisfies h^∗_i = h^∗_d−i for i ∈ {0, 1, 2, ..., d}.

Example 2.2.12. Going back to Example 2.1.2, we proved that for the unit cube d, Ehr_d(z) =X

t≥0

(t + 1)^dz^t.

This can be rewritten as

Ehr_d(z) =X

t≥0

(t + 1)^dz^t= Pd

k=0A(d, k)z^k (1 − z)^d+1 ,

where A(d, k) is the Eulerian number (they are often defined in terms of the identity above). So the h^∗-polynomial of ^d is

h^∗_

d(z) = Ad(z) :=

d

X

k=0

A(d, k)z^k.

Eulerian numbers have many other properties. Most importantly, they count certain statistics over permu- tations. For a permutation π = π1π2...πd, a descent is an index i where 1 ≤ i < d such that πi > πi+1. Now, it turns out that A(d, k) is the number of permutations of length d with exactly k descents. In other words,

Ad(z) =

d

X

k=0

A(d, k)z^k = X

π∈Sd

z^des(π), where des(π) is the number of descents in π.

2.3 Distributional properties of h

-polynomials

In the previous section, we saw that h^∗-polynomials for lattice simplices count the number of lattice points on different heights of the fundamental parallelepiped. For general lattice polytopes, it is a bit more un- clear what the h^∗-polynomials count, but Stanley’s non-negativity theorem seems to imply that they count something. In the case of the unit cube, the h^∗-polynomial turned out to count the number of permutations with a certain number of descents. There are many other such distributions that can also be described as an h^∗-polynomial of some polytope. Therefore, we are interested in their distributional properties, such as unimodality and symmetry.

A polynomial

p(x) = p₀+ p₁x + p₂x²+ ... + p_nxⁿ

is unimodal if p0 ≤ p1 ≤ ... ≤ pm ≥ pm+1 ≥ ... ≥ pn for some m ∈ {0, 1, 2, ..., n}. It is log-concave if p²_i ≥ pi−1pi+1 for every i ∈ {1, 2, ..., n − 1}. It is symmetric with respect to d if pi = pd−i for every

i ∈ {0, 1, ..., d}. A polynomial that is symmetric with respect to its degree is sometimes just said to be sym- metric. Note that by Hibi’s theorem (Theorem 2.2.11), the h^∗-polynomial of a reflexive polytope is always symmetric. A polynomial with real coefficients is said to be real-rooted if all its roots are real, or if it is identically zero.

For a polynomial p(x) = p₀+ p₁x + · · · + p_nxⁿ, an internal zero is an index i such that p_i = 0 and there exists numbers j < i and k > i where p_j 6= 0 and p_k 6= 0. Unimodality and log-concavity are related to real-rootedness in the following way:

Theorem 2.3.1 (Theorem 1.2.1 in [9]). If p(x) = p0+ p1x + p2x²+ · · · + pnxⁿ is real-rooted with nonnegative coefficients, then it is log-concave with no internal zeros which implies that it is unimodal.

The following are some other first results that are useful in determining when a given polynomial is symmetric and/or unimodal.

The mode of a unimodal polynomial p(x) = p₀+ p₁x + ... + p_nxⁿ is the index i such that p_i is maximal.

If there are multiple such indices, then the mode is defined as the median of them. Note that the mode of a symmetric and unimodal polynomial is always d/2 where d is the degree.

Proposition 2.3.2. The sum of two unimodal polynomials p(x) and q(x) with the same mode is unimodal.

Proof. Let m be the mode. Note that p₀ ≤ p1 ≤ ... ≤ p_bmc ≥ p_bmc+1 ≥ ... ≥ pn and the same is true for q(x), so

p₀+ q₀≤ p₁+ q₁≤ ... ≤ p_bmc+ q_bmc≥ p_bmc+1+ q_bmc+1≥ ... ≥ p_n+ q_n. This proves that p(x) + q(x) is unimodal.

Corollary 2.3.3. The sum of a finite number of unimodal polynomials that are symmetric with respect to d, is also unimodal and symmetric with respect to d.

Proposition 2.3.4 (Theorem 1 in [1]). Let

p(x) = p0+ p1x + ... + pnxⁿ and

q(x) = q0+ q1x + ... + qmx^m

be two unimodal and symmetric polynomials. Then p(x)q(x) is also unimodal and symmetric.

In Chapter 5, we will use these various results in order to prove that a certain family of h^∗-polynomials are always symmetric and unimodal. To do so we will prove first in Chapter 4 that a family of box polynomials are always real-rooted. Box polynomials and their connection to unimodality of h^∗-polynomials is the topic of the next section.

2.4 Triangulations and Box Polynomials

In this section, box polynomials and some applications of them will be presented. Box polynomials will be featured heavily in the following chapters, especially Chapter 4. So let us start by defining them:

Definition 2.4.1. Let P be a d-dimensional lattice simplex, and Π^◦ the interior of its fundamental paral- lelepiped. The box polynomial B_P(z) is defined as

B_P(z) := σΠ^◦(1, 1, 1, ..., 1, z) =

d

X

k=0

bkz^k,

where bk is the number of lattice points of Π^◦ with last coordinate k.

One useful property of box polynomials is that they are always symmetric.

Proposition 2.4.2. For any d-dimensional lattice simplex P, B_P(z) is symmetric with respect to d + 1.

Proof. Let

B_P(z) =

d

X

i=0

b_izⁱ.

To prove that b_k= b_d+1−k, we will find a bijection between the subset of Π^◦_P∩ Z^d+1 with last coordinate k, and the subset with last coordinate d + 1 − k. Pick a point

w =

d+1

X

i=1

λ_iw_i

in Π^◦_P∩ Z^d+1 with last coordinate k, and map it to

f (w) :=

d+1

X

i=1

(1 − λ_i)w_i.

Since x was in the interior of Π_P, the weights λ_i are strictly between 0 and 1, so f (w) will also be in Π^◦_P∩ Z^d+1, but it will have last coordinate d + 1 − k. Also, f is a bijection since we can invert it by applying it a second time.

The main application of box polynomials is a theorem by Betke and McMullen that gives a way to express h^∗-polynomials as a kind of weighted sum of certain box polynomials. Before stating that theorem, we will need some more definitions.

A simplicial polytope is a convex polytope whose proper faces (faces other than the polytope itself) are all simplices. The faces of a simplicial polytope can be thought of as a triangulation of its boundary (recall from Definition 2.1.1 that triangulations are defined for any set S ⊆ Rⁿ).

Let S ⊆ Rⁿ, and T a triangulation of S. Define the f -polynomial f (T ; z) of T as

f (T ; z) :=

d

X

i=0

fi−1zⁱ,

where fiis the number of i-dimensional simplices of T if i ≥ 0, and f₋₁:= 1. Similarly, let the h-polynomial of T , h(T ; z) = h0+ h1z + h2z²+ ... + hdz^d, be defined by

d

X

i=0

f_i−1(z − 1)^d−i=

d

X

i=0

h_iz^d−i.

Let S ⊆ Rⁿ, and T a triangulation of S. For a simplex ∆ ∈ T , define linkT(∆) to be the set of simplices in T that are disjoint from ∆, but are contained in a simplex of T that also contains ∆. In other words,

linkT(∆) := {Ω ∈ T such that Ω ∩ ∆ = ∅ and there exists ∆⁰∈ T such that ∆, Ω ⊂ ∆⁰}

Note that linkT(∆) is itself a triangulation, and so it has an f -polynomial and an h-polynomial.

Theorem 2.4.3 (Betke-McMullen). Let P be a lattice d-polytope, and T a triangulation of P. Then h^∗_P(z) = X

∆∈T

h(link_T(∆); z)B_∆(z).

If we have a reflexive polytope, then the following corollary to Theorem 2.4.3 (which follows directly from Theorem 10.5 in [7]) can be used instead.

Corollary 2.4.4 (Theorem 10.5 in [7]). Let P be a reflexive lattice d-polytope and T a triangulation of its boundary. Then

h^∗_P(z) = X

∆∈T

h(linkT(∆); z)B∆(z).

This corollary will be especially useful in Chapter 5. It can be used to prove unimodality and symmetry for h^∗-polynomials under certain conditions. First, we need the polynomials h(linkT(∆); z) to be unimodal and symmetric, and have degree d − dim(∆) − 1. Since the box polynomials are symmetric with respect to dim(∆) + 1, this would imply that the product h(linkT(∆); z)B∆(z) is symmetric with respect to d.

Therefore, the sum of them would be symmetric with respect to d as well. If, in addition to this, the box polynomials are unimodal, then by Proposition 2.3.4 this would imply that h(linkT(∆); z)B∆(z) is unimodal and symmetric. This would in turn mean that their mode is d/2, and since the sum of unimodal polynomials with the same mode is unimodal (see Corollary 2.3.3), this would mean that h^∗_P(z) is unimodal. This is why we are interested in proving unimodality for box polynomials. For boundaries of simplicial polytopes, the conditions needed above on the h-polynomials are satisfied. This is true thanks to the following lemma, which follows from Lemma 2.9 in [27].

Lemma 2.4.5 (Lemma 2.9 in [27]). Let T be the boundary of a simplicial polytope and let ∆ ∈ T . Then h(link_T(∆); z) has degree d − dim(∆) − 1, is symmetric, and is unimodal.

On the other hand, the following lemma says that if the triangulation is regular, then the polytope does not have to be simplicial.

Lemma 2.4.6 (Lemma 9 in [10]). Let P be a polytope with a regular triangulation that induces a triangulation T on the boundary. Then there exists a simplicial polytope Q such that the boundary triangulation T⁰ of Q is combinatorially equivalent to T . In particular, for each ∆ ∈ T ,

h(linkT(∆); z) = h(linkT⁰(∆⁰); z) for some ∆⁰∈ T⁰.

These applications of box polynomials to unimodality of h^∗-polynomials were first stated in [24], where the authors collected these various necessary properties into a definition.

Definition 2.4.7. A triangulation T of a lattice polytope P is called box unimodal if 1. T is regular, and

2. The box polynomials of all ∆ ∈ T are all unimodal.

Note that thanks to Theorem 2.4.3 and the arguments above, a reflexive polytope that admits a box unimodal triangulation has a unimodal and symmetric h^∗-polynomial.

Chapter 3

s-Eulerian Polynomials

In this chapter, a generalization of Eulerian polynomials, called the s-Eulerian polynomials, will be pre- sented. In recent years, the s-Eulerian polynomials have been the focus of numerous research projects in combinatorics (see for example [5, 6, 12, 17, 21, 22, 23]). They turn out to be the h^∗-polynomial of a certain kind of lattice simplex, and it is the box polynomials of these polytopes that are studied in this thesis. As a special case of these polynomials, we will recover the regular Eulerian polynomials. Recall that the Eulerian polynomials count descents in permutations. In order to relate these to the lattice points counted by the h^∗-polynomial, we will need a different representation of permutations, called Lehmer codes.

3.1 Lehmer codes

There are n! permutations of length n. How can we represent each permutation uniquely as a number from 0 to n! − 1 in a good way? Lehmer codes provide an answer to that question.

Definition 3.1.1. For a permutation π = π1π2...πn, define the Lehmer code for π as L(π) := (l1, l2, ..., ln),

where

l_i= #j ∈ {i + 1, i + 2, ..., n} such that π_i > π_j.

Let hni denote the set {0, 1, · · · , n − 1}. L(π) is an element in hni × hn − 1i × ... × h1i. We can also map from Lehmer codes to permutations, thereby proving that the map L is a bijection.

Proposition 3.1.2. The map L : Sn→ hni × hn − 1i × ... × h1i is a bijection.

Proof. In order to invert the map L, take a Lehmer code l = (l1, l2, ..., ln). First, create a set of numbers A = {1, 2, ..., n} and a (currently empty) permutation p = ∅. Go through l from left to right. For each li, take the li+ 1:th smallest number from A, remove it from A, and append it to p. After this is done, p will be the unique permutation such that L(p) = l.

To answer the question posed at the beginning of the section, we can get a number from a Lehmer code like this:

f (l) = (n − 1)!l₁+ (n − 2)!l₂+ ... + 0!l_n.

This number also has the nice property that f (L(π)) < f (L(σ)) if and only if π is lexicographically smaller than σ. More important to us is the following property. Recall that in Example 2.2.12 we defined the descents of a permutation. There is an easy way to spot descents by looking at the Lehmer code:

Lemma 3.1.3. Let π ∈ Sn and L(π) = l1l2...ln. Then i is a descent of π if and only if li> li+1.

Proof. Assume that i is a descent. Since πi > πi+1, for any index j > i + 1 such that πi+1 > πj we will also have that πi > πj, so li≥ li+1. However, since πi > πi+1, li will increase by one, so li> li+1. For the other direction, assume that l_i > l_i+1. If i was not a descent, then for any j > i such that π_i> π_j we would also have π_i+1> π_j. This means that l_i≤ li+1, which is a contradiction.

Lehmer codes give an alternative way of describing permutations, and this will help us in connecting permutations to the objects appearing in the following sections.

3.2 s-inversion sequences

In this section, a kind of generalization of Lehmer codes, called s-inversion sequences, will be introduced.

Definition 3.2.1. Let s = (s₁, s₂, ..., s_n) be a sequence of positive integers. An s-inversion sequence is a sequence of non-negative integers e = (e₁, e₂, ..., e_n) such that 0 ≤ e_i < s_i for all i = 1, 2, ..., n. The set of s-inversion sequences is denoted Is.

Just like for permutations, we can define ascents and descents of s-inversion sequences.

Definition 3.2.2. For an s-inversion sequence e, the number of ascents is defined as asc(e) := #i ∈ {0, 1, 2, ..., n − 1} such that ei

si

< ei+1

si+1

,

where s₀:= 1. Similarly, the number of descents is defined as des(e) := #i ∈ {1, 2, ..., n} such that ei

s_i >ei+1

s_i+1, where sn+1:= 1.

In other words, the number of ascents (descents) of an s-inversion sequence is the number of pairs of adjacent elements where ^e_sⁱ

i < ^e_sⁱ⁺¹

i+1

e_i s_i > ^e_sⁱ⁺¹

i+1



, except that you get one extra ascent (descent) if the first (last) element is non-zero. For reasons that will become more clear later, this turns out to be the most

“natural” definition of ascents (descents). We can think of the “extra” (i.e. s₀ = 1 and s_n+1= 1) part as adding a 1 to the beginning and end of s. This leads to the following alternative definition.

Definition 3.2.3 (Alternative to Definition 3.2.2). For an s-inversion sequence e, let s⁰ = (1, s, 1), and let e⁰ be the s⁰-inversion sequence (0, e, 0). The number of ascents of e is defined as the number of adjacent elements in e⁰ where

e⁰_i s⁰_i <e⁰_i+1

s⁰_i+1. The number of descents is defined similarly.

To recover the regular Eulerian polynomials, the following choice of s is used:

Proposition 3.2.4. Let s = (1, 2, 3, ..., n). Then X

e∈Is

z^asc(e)= X

π∈Sn

z^des(π).

Proof. We will find a bijection f : I_s → Sn such that des(f (e)) = asc(e). For some e ∈ I_s, let f (e) be the permutation corresponding to the Lehmer code

l_e:= e_ne_n−1...e₁.

Figure 3.1: An s-inversion sequence with two ascents and two descents. The bars represent s and the dots represent e.

Thanks to Proposition 3.1.2, this is a bijection. Also, since e1is always 0, we will not have any extra ascents in e. An ascent in e is a pair of adjacent elements such that

ei

si

< ei+1

si+1

= ei+1

si+ 1. This simplifies to

s_ie_i+ e_i< s_ie_i+1 ⇐⇒ ei< e_i+1.

However, in l_e, e_i < e_i+1 corresponds to a pair of adjacent elements where l_j > l_j+1. Due to Lemma 3.1.3, this corresponds to a descent in f (e). So the proof is done.

Recall from Example 2.2.12 that P

π∈Snz^des(π) is the Eulerian polynomial An(z). For this reason, the polynomial

Es(z) := X

e∈I_s

z^asc(e)

is called the s-Eulerian polynomial. For the regular Eulerian polynomials, it does not matter whether we use des(π) or asc(π) because they are equidistributed:

An(z) = X

π∈Sn

z^des(π) = X

π∈Sn

z^asc(π).

The reason for this is that each permutation with k descents and l ascents can be mapped bijectively to a permutation with k ascents and l descents by swapping each π_i with n − π_i+ 1 (another way is to reverse π). This is actually true for s-Eulerian polynomials as well:

Theorem 3.2.5. For any sequence of positive integers s, Es(z) =X

e∈I_s

z^asc(e)= X

e∈I_s

z^des(e).

Proof. Just like with the regular Eulerian polynomials, the idea is to find a bijection from Is that maps elements with k ascents and l descents to elements with k descents and l ascents. The bijection looks like this:

f (e) := −e mod s,

meaning that the i:th element of f (e) is −ei modulo si. So, for example, if s = (5, 4, 5) and e = (0, 3, 1) then f (e) = (0, 1, 4). This is a bijection, because we can invert it by just applying f again. Replacing ei

with −ei mod si will “turn around” the numbers, meaning that ascents should turn into descents and vice versa. Unfortunately, that is not always the case, because zeros get mapped to themselves. However, it will work out anyway since we fixed two zeros at either end of e⁰ in Definition 3.2.3. To be a bit more precise: if e_i> 0, then f (e)_i= s_i− e_i, and so ^{f (e)}_s ⁱ

i = 1 −^e_sⁱ

i. This implies that if e_i> 0 and e_i+1> 0, then 1. If ^e_sⁱ

i < ^e_sⁱ⁺¹

i+1 then ^{f (e)}_s ⁱ

i > ^{f (e)}_s ⁱ⁺¹

i+1 , 2. If ^e_sⁱ

i > ^e_sⁱ⁺¹

i+1 then ^{f (e)}_s ⁱ

i < ^{f (e)}_s ⁱ⁺¹

i+1 , and 3. If ^e_sⁱ

i = ^e_sⁱ⁺¹

i+1 then ^{f (e)}_s ⁱ

i = ^{f (e)}_s ⁱ⁺¹

i+1 .

Let a_ebe the number of ascents between non-zero numbers of e⁰= (0, e, 0), and d_eis the number of descents between non-zero numbers. What we proved above is that a_e = d_{f (e)} and d_e = a_{f (e)}. There is one more way an ascent can occur: if e⁰_i = 0 and e⁰_i+1 > 0. Let αe be the number of such ascents, and let δe be the number of descents where e⁰_i> 0 and e⁰_i+1= 0. Now since e⁰ has a zero on both endpoints, both αe and δe

are equal to the number of contiguous segments of non-zero numbers in e⁰, and in particular, αe= δe. Also, since f maps zeros to zeros and non-zeros to non-zeros, it holds that αe= α_{f (e)}and δe = δ_{f (e)}. Combining all this, we get

des(e) = de+ δe= a_{f (e)}+ α_{f (e)}= asc(f (e)), and

asc(e) = a_e+ α_e= d_{f (e)}+ δ_{f (e)}= des(f (e)).

This proves the statement.

There are a lot of well-studied polynomials in combinatorics that can be written as s-Eulerian polyno- mials, so proving things about them is important. As noted in Chapter 2, there is an avid interest in the distributional properties of polynomials whose coefficients form a combinatorial sequence. Thus, Savage and Visontai [23] proved that they are real-rooted, and therefore unimodal.

Theorem 3.2.6 (Theorem 1.1 in [23]). For any sequence of positive integers s, the s-Eulerian polynomial Es(z) is real-rooted.

3.3 s-lecture hall simplices

As mentioned at the start of Chapter 3, the s-Eulerian polynomials are actually the h^∗-polynomials of certain lattice simplices.

Definition 3.3.1. For a sequence s of positive integers, the s-lecture hall simplex Ps is defined as Ps:=



(x₁, ..., x_n) ∈ Rⁿ| 0 ≤ x₁ s1

≤ ... ≤ x_n sn

≤ 1

 . This can also be written in terms of the vertices of Ps:

P_s= conv



















 0 0 ... 0









 ,









 0 0 ... sn









 ,









 0

... sn−1

sn









 , ...,









 s1

s2

... sn



















 .

Figure 3.2: The s-lecture hall simplex Ps, where s = (2, 3).

The s-lecture hall simplices are connected to the s-inversion sequences in the following way:

Theorem 3.3.2 (Theorem 3.9 in [18]). Let s be a sequence of positive integers. Then the s-Eulerian polynomial is the h^∗-polynomial of P_s.

This was first proved in [22]. In [18], there is a proof that uses a certain bijection between s-inversion sequences and the fundamental parallelepiped of Ps, which we will denote by Πs. This bijection will be useful later, so we state it here along with a somewhat sketchy version of the proof:

Proof of Theorem 3.3.2. The idea is to find a bijection between Πs∩ Zⁿ and Is, such that elements with last coordinate k get mapped to elements with k descents. Recall from Proposition 2.2.8 that the coefficients of the h^∗-polynomial count the number of lattice points of Π_s∩ Zⁿ with different last coordinates, so this kind of bijection would prove the theorem. First, let’s examine what elements in Π_s∩ Zⁿ look like. Recall that

Πs = {λ1w1+ ... + λnwn such that all 0 < λi≤ 1}.

Let x ∈ Π_s∩ Zⁿ. Then

x =







 x1

x2

...

x_n+1









=









λns1

(λn−1+ λn)s2

...

(λ₀+ ... + λ_n)







 .

Note that λn= ^x_s¹

1. Thus,

0 ≤ x1

s1

< 1, and xi+1/si+1− xi/si= λn−i for i ∈ {1, 2, ..., n − 1}. Therefore,

0 ≤ x_i+1 si+1

−x_i si

< 1.

Finally, λ0= xn+1−^x_sⁿ

n, and so

0 ≤ xn+1−x_n sn

< 1.

However, this just means that xn+1 is always equal to d^x_sⁿ

ne. In fact, any x satisfying the three constraints above will be in Πs∩ Zⁿbecause we can recover the λi:s by taking the differences xi+1/si+1− xi/si(they are

Figure 3.3: A graphical representation of the REM-function. Notice that the last coordinate of x is 3, which is also the number of descents of REM(x).

the hyperplane description of Πs). If we ignore the last element of x, it looks a little bit like an s-inversion sequence already, but there are two main differences. First, ^x_sⁱ

i ≤ ^x_sⁱ⁺¹

i+1, so it is “weakly increasing” and has no descents. Also, there is no guarantee that ^x_sⁱ

i < 1. So xi can go “outside of si”. Therefore, the way to get a bijection to Is is to ignore the last coordinate of x, and take the other elements modulo si. Let REM: Πs∩ Zⁿ→ Is be the function

REM(x) := (xi mod si)ⁿ_i=1.

Here is some intuition as to why the last coordinate of x ends up being the number of descents of REM(x): For each i ∈ {1, 2, 3, ..., n}, we can consider the number b^x_sⁱ

ic as some kind of “winding number”. If b^x_sⁱ

ic = b^x_sⁱ⁺¹

i+1c then since ^x_sⁱ

i ≤ ^x_sⁱ⁺¹

i+1, it will not give a descent in REM(e). If, on the other hand, b^x_sⁱ

ic = b^x_sⁱ⁺¹

i+1c − 1, then since ^x_sⁱ⁺¹

i+1 −^x_sⁱ

i < 1, this will always give a descent in REM(e). Therefore, the number of descents should be equal to the “total winding number” b^x_sⁿ

nc = bλ1+ ... + λnc. This is equal to xn+1 if and only if λ0= 0, which happens if and only if REM(x)n= 0. Otherwise, xn+1= bλ1+ ... + λnc + 1 . However, according to the definition of descents, we should get one extra descent when e_n> 0. So this works out nicely.

Since P_s is a simplex, it also has an associated box polynomial. These box polynomials turn out to generalize interesting combinatorial polynomials. This is the topic of the next chapter.

Chapter 4

s-Derangement Polynomials

In this chapter, we will investigate the box polynomials (see Definition 2.4.1) of s-lecture hall simplices. In particular, we would like to prove that they generalize the well-studied derangement polynomials, and that they are real-rooted. Before we can do that, we must translate the open parallelepiped Π^◦_s to s-inversion sequences using the “REM”-function from the proof of Theorem 3.3.2.

4.1 The box polynomial of an s-lecture hall polytope

For a sequence of positive integers s, let Bs(z) denote the box polynomial of Ps. Proposition 4.1.1. For a sequence of positive integers s,

Bs(z) =X

e∈I_s

z^asc(e)=X

e∈I_s

z^des(e),

where Is is the subset of Is where e16= 0, en6= 0, and ^e_sⁱ

i 6=^e_sⁱ⁺¹

i+1 for i ∈ {1, 2, ..., n − 1}.

As we go through the proof, the definition of Is will make more sense, and an alternative definition, similar to Definition 3.2.3, will be presented.

Proof of Proposition 4.1.1. The open parallelepiped Π^◦_s is a subset of Π_s. So if we just map Π^◦_s to s-inversion sequences using the REM-function, we will get an identity like the one above for some subset I_s⊆ I_s. What remains to check is that this subset really is the one described above, and that ascents and descents are equidistributed on this set.

Recall that we found the hyperplane description of Πs in the proof of Theorem 3.3.2. Since Π^◦_s is the interior of Πs, it has the same hyperplane description except that all inequalities have turned strict. So Π^◦_s∩ Zⁿ is given by the vectors x ∈ Πs∩ Zⁿ that also satisfy:

1. 0 < ^x_s¹

1, 2. 0 < ^x_sⁱ⁺¹

i+1 −^x_sⁱ

i, for i ∈ {1, 2, ..., n − 1}, and 3. 0 < xn+1−^x_sⁿ

n.

In terms of the REM-function, ^x_s¹

1 > 0 means that x1∈ {1, 2, ..., s1− 1}, and so REM(x)16= 0. The second constraint means that ^x_sⁱ⁺¹

i+1 > ^x_sⁱ

i, and so ^REM(x)_s ⁱ

i 6= ^REM(x)_s ⁱ⁺¹

i+1 . Recall that xn+1 is always given by d^x_sⁿ

ne.

Thus, the third constraint is equivalent to ^x_sⁿ

n < d^x_sⁿ

ne. Equivalently, s_n6 | xn,