University of Kentucky Discrete CATS Seminar
U N I V E R S I T Y   O F   K E N T U C K Y


2017 -- 2018

Fall 2017

Unless otherwise noted, seminar meets at 2:00 pm Mondays in 745 POT.

Sept 18 Richard Ehrenborg
University of Kentucky
Simion's type B associahedron is a pulling triangulation of the Legendre polytope
We show that the Simion type B associahedron is combinatorially equivalent to a pulling triangulation of the type A root polytope known as the Legendre polytope. Furthermore, we show that every pulling triangulation of the boundary of the Legendre polytope yields a flag complex. Our triangulation refines a decomposition of the boundary of the Legendre polytope given by Cho.

This is joint work with Gábor Hetyei and Margaret Readdy.

Sept 25 Zhexiu Tu
Centre College
Topological Representations of Matroids and the cd-index
There are several different topological representations of non-orientable matroids. In this talk, inspired by Swartz's work, I will show an explicit fully partitioned homotopy sphere d-arrangement S that is a CW-complex whose intersection lattice is the geometric lattice of the corresponding matroid for matroids of rank < 5. Moreover S has a d-sphere in it that is a regular CW-complex. We will also look at enumerative properties, including how the flag f-vector formula of Billera, Ehrenborg and Readdy for oriented matroids applies to arbitrary matroids.
Oct 2 Alex Happ
University of Kentucky
The sum of powers of the descent set statistic
We study the sum of the rth powers of the descent set statistic and how many small prime factors occur in these numbers. The results will depend upon the base p expansion of n and r.

This is joint work with Richard Ehrenborg

Oct 9 McCabe Olsen
University of Kentucky
Level algebras and lecture hall polytopes
Given a family of lattice polytopes, a common question in Ehrhart theory is classifying the which polytopes in the family are Gorenstein. A less common question is classifying which polytopes in the family admit level semigroup algebras, a generalization of the Gorenstein property. In this talk, we consider these questions for lecture hall polytopes. We provide a characterization of the Gorenstein property for a large subfamily of lecture hall polytopes. Additionally, we also provide a complete characterization for the level property.

This is joint work with Florian Kohl

Oct 30 Rafael González D'león
Universidad Sergio Arboleda
and York University
The Whitney dual of a graded poset
Two posets are Whitney duals to each other if the (absolute value of their) Whitney numbers of the first and second kind are switched between the two posets. We introduce new types of edge and chain-edge labelings of a graded poset which we call Whitney labelings. We prove that every graded poset with a Whitney labeling has a Whitney dual and we show how to explicitly construct a Whitney dual using a technique that involves quotient posets. As an application of our main theorem, we show that geometric lattices, the lattice of noncrossing partitions, the poset of weighted partitions studied by González D'León-Wachs and the R*S-labelable posets studied by Simion-Stanley all have Whitney duals. We also show that a graded poset P with a Whitney labeling admits a local action of the 0-Hecke algebra on the set of maximal chains of P. The characteristic of the associated representation is Ehrenborg's flag quasisymmetric function of P. This is joint work with Josh Hallam (Wake Forest Universtity).
Nov 6 Megan Bernstein
Georgia Tech
Progress in showing cutoff for random walks on the symmetric group
Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed distribution. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk, generated by permutations with a binomially distributed number of two-cycles. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group.

Includes joint work with Nayantara Bhatnagar, Evita Nestoridi, and Igor Pak.

Nov 13 Radmila Sazdanovic
NC State University
Chromatic homology theories
This talk is an entree to categorification through knot theory and graph theory. The focal point is the chromatic polynomial and is categorifications: chromatic graph homology over algebra defined by L. Helme-Guizon and Y. Rong, and the homology of a graph configuration space introduced by M. Eastwood, S. Huggett. Time permitting, we will discuss relations between these homology theories in the form of spectral sequences, as well as a new invariant of simplicial complexes inspired by the Eastwood and Huggett approach.
Nov 20 Andy Wilson
U Penn
The combinatorics of symmetric quotient rings
The coinvariant ring of the symmetric group is the quotient of the polynomial ring by the ideal generated by all symmetric polynomials without a constant term. Many properties of this ring are closely connected to the combinatorics of the symmetric group. What if, instead, we mod out by an ideal generated by some other set of polynomials? If the ideal is symmetric, can we use combinatorics to understand the properties of the resulting quotient ring? A variety of authors (Rhoades, Haglund, Shimozono, Huang, Scrimshaw, the speaker, and others) have discovered many well-behaved quotient rings this way. Furthermore, they have shown that the rings are connected to classical combinatorial objects like ordered set partitions and words. We will provide an overview of the work in this area and pose a conjecture that, if proven, would unify much of the existing work on this problem.
Nov 27 Joseph Cummings
University of Kentucky
Cohen-Macaulay Stanley-Reisner Rings
It turns out that in order to solve many purely combinatorial problems relating to simplicial complexes, one needs to study algebraic properties of its Stanley-Reisner ring. For example, we will show the Cohen-Macaulay condition gives us sharp bounds on the complex’s h-vector. We will also discuss Reisner’s criterion which gives an equivalent combinatorial criterion for Cohen-Macaulyness, and time permitting, Stanley’s upper bound theorem for simplicial spheres.
Dec 4 Marie Meyer
University of Kentucky
Polytopes Associated to Graphs
There are many advantageous ways to associate a polytope to a graph. In this talk, we will discuss a couple of such constructions while highlighting some notable results. First we will look at edge polytopes introduced by Ohsugi and Hibi as well as applications through constructive examples from a paper by Lason and Michalek. Then we will look at Laplacian simplices associated to graphs and digraphs in the time remaining.

Spring 2018

Jan 22 Brian Davis
University of Kentucky
Regular triangulations and Gröbner bases
In this talk we will give a friendly introduction to regular triangulations, which is a tool for breaking down an integer polytope into simpler pieces: high dimensional triangles! In the second part of the talk we will present a context in which triangulations make a normally difficult computation much easier. The main theorem, presented without proof, is really charming! We assume no knowledge of commutative algebra beyond the prelim sequence.
Jan 29 McCabe Olsen
University of Kentucky
Ehrhart theory and ordered set partitions
Given a lattice polytope P, one of the central questions in Ehrhart theory is to describe the h*-polynomial (or h*-vector) of P, as this encodes and detects certain algebraic and geometric properties of P. Given that the coefficients of the h*-polynomial are nonnegative integers, it is natural (for a combinatorialist) to wishfully think that these coefficients count something; that is, ideally this polynomial encodes some statistic on some combinatorial object. In this talk, we will discuss some conjectures of Nick Early regarding the h*-polynomial of two well-known polytopes, namely dilated unit simplices and hypersimplices, involving a winding number statistic on certain decorated ordered set partitions. We will provide a proof to one of these conjectures and discuss the other.
Feb 5 Ben Braun
University of Kentucky
Ehrhart h* polynomials, unit circle roots, and Ehrhart positivity
We will introduce the basics of Ehrhart theory, then discuss methods for establishing Ehrhart positivity using h* polynomials with roots on the unit circle. This is based on joint work with Fu Liu.
Feb 12 Karthik Chandrasekhar
University of Kentucky
Facing up to metrics
Here we study arrangements of geodesic lines on spaces topologically equivalent to the real 2-plane, but having different metrics. We study the regions of all possible f-vectors. Mainly we study the hyperbolic plane (the upper half plane with a non-Euclidean metric) which reveals a vastly different region of f-vectors.
Feb 19 Carl Lee
University of Kentucky
The Ingredients of the g-Theorem
I will present some of the historical context of the g-Theorem, which characterizes the numbers of faces of simplicial convex polytopes, and the ingredients of its proof.
Feb 26 Andrés R. Vindas Meléndez
University of Kentucky
Fixed Subpolytopes of the Permutahedron
Motivated by the generalization of Ehrhart theory with group actions, this project makes progress towards obtaining the equivariant Ehrhart theory of the permutahedron. The fixed subpolytopes of the permutahedron are the polytopes that are fixed by acting on the permutahedron by a permutation. We prove some general results about the fixed subpolytopes. In particular, we compute their dimension, show that they are combinatorially equivalent to permutahedra, provide hyperplane and vertex descriptions, and prove that they are zonotopes. Lastly, we obtain a formula for the volume of these fixed subpolytopes, which is a generalization of Richard Stanley's result of the volume for the standard permutahedron. This is joint work with Federico Ardila (San Francisco State) and Anna Schindler (University of Washington).
Mar 5 Gábor Hetyei
UNC Charlotte
Partitions of a fixed genus have an algebraic generating function
We show that, for any fixed genus g, the ordinary generating function for the genus g partitions of an n-element set into k blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus 2 partitions, after identifying all genus 2 primitive partitions, using a computer-assisted search. This is joint work with Robert Cori.
Mar 12 Spring Break

Mar 19 Nick Early
University of Minnesota
Canonical bases for permutohedral plates
There is a natural construction according to which the set of all faces of an arrangement of hyperplanes can be made into a vector space, by taking linear combinations of their characteristic functions. Our space is equipped with a natural basis of characteristic functions of certain polyhedral cones called permutohedral cones passing through the origin, studied as plates by A. Ocneanu, which are labeled by ordered set partitions; these are in duality with faces of the arrangement of reflection hyperplanes xi=xj. It is interesting to study the affine case as well: recently we conjectured stating that plates situated compatibly with the lattice of integer points in a generalized hypersimplex provide a new combinatorial interpretation of the h*-vector for that generalized hypersimplex.

In this talk, we construct directly a certain canonical basis which is compatible with one or both of two quotients: neglecting characteristic functions of (1) nonpointed cones, and (2) cones of codimension at least 1. The essential feature here is that subsets of the canonical basis map to bases of the quotients. As a consequence, we obtain the straightening relations which were originally computed by Ocneanu through the introduction of an auxiliary space of formal linear combinations of layered trees. Time permitting, we will describe the circumstances which led us to formulate the conjecture for the h*-vector.

Apr 2 Aida Maraj
University of Kentucky
Hierarchical models and their toric ideals
This talk will be about Hierarchical Models in Algebraic Statistics. I will present an example where these models apply and introduce their corresponding toric ideals. A formula for calculating Krull dimension will be given. To describe generating sets of these ideals one can use a symmetric group action. Using this tool, we will describe generating sets for some classes of these ideals as decomposable and non-reducible Models. This is joint work with Uwe Nagel.
Apr 3 McCabe Olsen University of Kentucky
PhD Defense
CB 212, 3-4 pm
Apr 4 Jessica Doering
University of Kentucky
Masters Exam
745 POT, 1 pm
Apr 5 Alex Happ
University of Kentucky
PhD Defense
318 POT, 10 am
A combinatorial miscellany: antipodes, parking cars and descent set powers
Advisor: Richard Ehrenborg
Apr 10 Kyle Franz
University of Kentucky
Masters Exam
10 am
Apr 12 Marie Meyer
University of Kentucky
PhD Defense
103 Chemistry Physics, 11 am
Apr 16 Martha Yip
University of Kentucky
The volume of the caracol polytope
A number of flow polytopes have volumes that are products of nice combinatorial quantities, but perhaps surprisingly, there are no known combinatorial proofs of these formulas. In this talk, I will give a combinatorial interpretation of the Lidskii volume formula of Baldoni and Vergne, and use this to prove that the volume of the caracol polytope is the product of a Catalan number and the number of parking functions of length n. This is joint work with Benedetti, Gonzalez D'Leon, Hanusa, Harris, Khare and Morales.
Apr 23 Open date

Seminar Archive

Margaret Readdy is the organizer of the Discrete CATS Seminar. If you wish to be added to the seminar list or give a talk, please e-mail her at margaret.readdy (at)

We would like to thank Benjamin Braun (NSA 2016--2018), Richard Ehrenborg (Simons Foundation Collaboration Grant 2016--2021), Margaret Readdy (Simons Foundation Collaboration Grant 2016--2021) and Martha Yip (Simons Foundation Collaboration Grant 2016--2021) who generously continue to support our speakers with their grants. The Discrete CATS seminar is also partially supported by the University of Kentucky Department of Mathematics.

Last updated April 15, 2018.