Discrete CATS Seminar
Department of Mathematics
University of Kentucky

Fall 2020

August 31 Andrés R. Vindas Meléndez (University of Kentucky)
Title: Decompositions of Ehrhart h*-polynomials for rational polytopes

Abstract: The Ehrhart quasipolynomial of a rational polytope P encodes the number of integer lattice points in dilates of P, and the h* -polynomial of P is the numerator of the accompanying generating function. We provide two decomposition formulas for the h*-polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our rational Betke--McMullen formula to provide a novel proof of Stanley's Monotonicity Theorem for the h*-polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the h*-polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes. This is joint work with Matthias Beck (San Francisco State Univ. & FU Berlin) and Ben Braun (Univ. of Kentucky)

September 7 Noah Speeter (University of Kentucky)
Title: Chip firings and the Scramble Number

Abstract: Chip firing can be described as a game we play by placing poker chips on the vertices of a graph. The scramble number is a new graph invariant that helps inform us as to how many chips we need to "win" that game. We will explore the properties of the scramble number as well as techniques used to compute it. This is joint work with Dave Jensen.

September 14 Mario Sanchez (UC Berkeley)
Title: The Universal Valuation of Coxeter Matroids

Abstract: Valuations on a family of polytopes are functions which behave nicely with respect to subdivisions in this family. One important question is the determine the structure of the set of all valuations on a certain family. This can be done by constructing a "universal valuation" which is a valuation that can be specialized to any other valuation on this family. Coxeter matroids are a generalization of matroids to an arbitrary root system. As with usual matroids, we can interpret Coxeter matroids as polytopes. In this talk, we will construct a universal valuation for the family of Coxeter matroid polytopes.

September 21 Emine Yildirim (Queens University)
Title: The Coxeter Transformation and Rowmotion for Cominuscule Posets

Abstract: There is a combinatorial action, called the Rowmotion, defined on cominuscule posets. It is well-known that this action has order 'h' on the order ideal poset of a cominuscule poset where h is the Coxeter number of the corresponding root system. Also, we will talk about the action of the Coxeter transformation on the order ideals of cominuscule posets. We prove that the Coxeter transformation is periodic of order 'h+1' (up to a sign) in most cases. We will demonstrate combinatorial similarities of the orbits of these two actions.

September 28 Gábor Hetyei (University of North Carolina - Charlotte)
Title: Rational Links Represented by Reduced Alternating Diagrams

Abstract: In knot theory, a rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a nonalternating form and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millet, but these rely on a special continued fraction expansion of the rational number in which all partial denominators are even (called all-even form}). In this talk we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced alternating form, or equivalently the nonalternating form of the corresponding rational number. This is joint work with Yuanan Diao and Claus Ernst. The talk will define and explain all terms, with a general audience in mind.

October 5 Véronique Bazier-Matte (University of Connecticut)
Title: Quasi-cluster algebras and triangulations of the Möbius strip

Abstract: In this talk, we will first define triangulations of marked surfaces, then use it to define quasi-cluster algebras (the equivalent of cluster algebras for non-orientable surfaces). We will list a few properties of these algebras, then we will count the number of triangulations of the Möbius strip, the only surface with a finite number of triangulations.

October 12 Greg Muller (University of Oklahoma)
Title: Linear recurrences indexed by Z

Abstract: We consider a system of equations in variables indexed by the integers, in which each variable is equal to a linear combination of the previous variables. We will show a number of general results about these systems, including an analog of Gaussian elimination, a parametrization of solutions, and (time-permitting) a characterization of systems whose solutions are periodic.

October 19 Ayomikun Adeniran (Pomona College)
Title: Increasing and Invariant Parking Sequences

Abstract: The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, the cars are allowed to have different sizes and each takes up a number of adjacent parking spaces after a trailer that was parked at the start of the street. A preference sequence in which all the cars are able to park is called a parking sequence. In this talk, we will look at increasing parking sequences and their connections to lattice paths. We will also discuss two notions of invariance in parking sequences and present various characterizations and enumerative results. This is joint work with Catherine Yan.

October 26 Liam Solus (KTH Royal Institute of Technology)
Title: Some Recent Applications of Real-rooted Polynomials

Abstract: In enumerative, geometric, algebraic and topological combinatorics the inequalities that hold amongst the coefficients of a combinatorial generating polynomial are frequently studied. Typical questions ask whether or not the coefficient sequence is unimodal, log-concave, alternatingly increasing and/or gamma-nonnegative. We will discuss some recent results that rely on the real zeros of polynomials to give answers to questions of this type. The main applications will pertain to polytopal cell complexes and lattice polytopes. Aside from giving answers, we will also pose some new problems motivated by these results.

November 2 Sara Billey (University of Washington)
Title: Limit Laws for q-Hook Formulas

Abstract: Various asymptotic aspects of the Hook Length Formula for standard Young tableaux have been studied recently in combinatorics and probability. In this talk, we study the limiting distributions that come from random variables associated to Stanley's q-hook-content formula for semistandard tableaux and q-hook length formulas of Björner--Wachs related to linear extensions of labeled forests. We show that, while these limiting distributions are "generically" asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the moduli space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded 2-norm. The closure of the moduli space of these distributions in the Lévy metric gives rise to the moduli space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box. This talk is based on joint work with Joshua Swanson at UCSD.

November 9 Richard Ehrenborg (University of Kentucky)
November 16 Carolina Benedetti (Universidad de los Andes)