DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2018 Box Polynomials of Lattice Simplices NILS GUSTAFSSON KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES 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´enand Leander regarding unimodality of h∗-polynomials of s-lecture hall order polytopes. 1 Sammanfattning Boxpolynomet av ett gittersimplex ¨aren variant av det mer k¨anda h∗-polynomet, d¨arden ¨oppnafunda- mentala parallelepipeden anv¨andsist¨alletf¨orden halv¨oppna.Boxpolynom ¨arkopplade till h∗-polynom tack vare en sats av Betke och McMullen fr˚an1985. Denna sats kan anv¨andasf¨oratt bevisa vissa egenskaper hos h∗-polynom, som t.ex. unimodalitet och symmetri. I denna uppsats unders¨oker vi boxpolynomen hos en s¨arskildfamilj av simplex, de s˚akallade s-h¨orssalssim- plexen. F¨ordessa simplex ¨ar h∗-polynomen en generalisering av de Eulerska polynomen, och visades ha endast reella r¨otterav Savage och Visontai 2015. Vi anv¨anderen modifierad version av deras bevis f¨oratt bevisa att ¨aven boxpolynomen bara har reella r¨otter,och att de ¨aren generalisering av derangemangpoly- nom. Vi anv¨andersedan dessa resultat f¨oratt delvis besvara en f¨ormodan av Br¨and´enoch Leander ang˚aende unimodaliteten hos h∗-polynomen av s-h¨orsalsordningspolytoper. 2 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 3 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`eneEhrhart 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 πi > πi+1. If we let the sequence fAn;0;An;1; ··· An;n−1g 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. 4 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. 5 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: n P = fλ1v1 + ::: + λmvm 2 R : λi ≥ 0 and λ1 + ::: + λm = 1g: The points v1; :::; vm are 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) = fx + λ(y − x): x; y 2 P; λ 2 Rg: 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.