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




2012 -- 2013

Spring 2013

Unless otherwise noted, seminar meets at 2:00 pm Mondays in 945 POT.

February 18 Elizabeth Niese
Marshall University

Factorizations of combinatorial Macdonald polynomials
The Hilbert series of the Garsia-Haiman module can be defined combinatorially as generating functions of certain fillings of Ferrers diagrams. One of the challenges in working with the combinatorial definition is the large number of fillings needed to generate a polynomial. In this talk we look at combinatorial proof of some factorizations of the Hilbert series.
March 4 Ben Braun
University of Kentucky
Chromatic polynomials: a survey
Chromatic polynomials are subtle invariants of finite, simple graphs. In this survey talk, we will discuss as many connections as time permits between chromatic polynomials and the following topics: deletion/contraction, Möbius inversion, characteristic polynomials of hyperplane arrangements, Ehrhart theory and inside-out polytopes, coloring complexes and Hilbert polynomials, Eulerian idempotents and Hodge decompositions, f-vectors of broken circuit complexes, and categorification via graded Euler characteristics of chain complexes.
March 11 No meeting Spring Break
March 18 Michelle Wachs
University of Miami
Special Event: Hayden-Howard Lecture

Eulerian Polynomials and Beyond
The Eulerian polynomials, which were introduced by Euler over 200 years ago, are pervasive throughout discrete mathematics and arise in a variety of surprising ways. Over the years generalizations and q-analogs possessing fundamental properties of the Eulerian polynomials have been introduced. I will discuss some of these generalizations and describe intriguing connections with algebraic geometry, representation theory and symmetric functions.

April 8 Sarah Nelson
University of Kentucky
QS-distribution, riffle shuffles, and quasisymmetric functions
In this talk, we will discuss part of Richard Stanley's paper "Generalized Riffle Shuffles and Quasisymmetric Functions". After defining the QS-distribution by standardizing elements from the probability distribution on a totally ordered set, we will examine another description in terms of riffle shuffles. Then we will consider the relationship between the QS-distribution and quasisymmetric functions. Using this relationship, we can generalize results concerning quasisymmetric functions and symmetric functions.

Masters Exam.
9:30 am 745 POT

Note change in time and place.

April 12 Cliff Taylor
University of Kentucky
Multitriangulations as Complexes of Star Polygons
A multitriangulation of order k, or a k-triangulation, of a convex n-gon is a maximal set of diagonals such that no k+1 of them mutually cross in the interior of the n-gon. First studied in the 1992 paper "A Turan-Type Theorem on the Chords of Convex Polygons" by Capoyleas and Pach, k-triangulations have recently been studied in the context of the multi-associahedron. In this talk, we will prove a result by Pilaud and Santos in the paper "Multi-triangulations as Complexes of Star Polygons", namely, that k-triangulations are formed by a union of k-stars and "k-irrelevant" edges. Time permitting, we will also discuss our recent work concerning the realization of the multi-associahedron.

Qualifying Exam.
2:00 pm 745 POT

Note change in day and place.

April 15 Robert Davis
University of Kentucky
Unimodality of h*-vectors
For a given lattice polytope, its Ehrhart series may be expressed as a rational function with nonnegative integer coefficients in the numerator. These coefficients, called the h*-vector of the polytope, can tell us about the polytope and the underlying algebraic structure. Of particular interest is determining when the h*-vector is unimodal. In this talk, we will discuss progress that has been made in this direction.
April 22 Bruce Sagan
Michigan State University

Factoring the Characteristic Polynomial of a Poset
Give a poset P, its characteristic polynomial χ(P,t) is the generating function in the variable t for the Möbius function of P. For many families of posets, every root of χ(P,t) is in the set of positive integers. A number of different techniques have been devised for showing that χ(P,t) factors over the positive integers, including Zaslavsky's theory of signed graphs, results by Saito and Terao about free hyperplane arrangements, and Stanley's Supersolvability Theorem. We will present a new, totally combinatorial method for proving factorization. This is joint work with Joshua Hallam.

Fall 2012

September 10 Margaret Readdy Organizational Meeting
Tea and cookies provided. BYOM = Bring your own mug.
September 17 Benjamin Braun
University of Kentucky
s-Lecture hall partitions, self-reciprocal polynomials, and Gorenstein algebras
In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions. We continue this line of investigation, connecting their work to the more general context of Gorenstein semigroup algebras. We focus on the Gorenstein condition for s-lecture hall cones when s is a sequence generated by a two-term recurrence with initial values 0 and 1. This is joint work with Matthias Beck, Matthias Koeppe, Carla Savage, and Zafeirakis Zafeirakopoulos.
September 24 Richard Ehrenborg
University of Kentucky
Prisms and pyramids of shelling components
We study how the cd-index of shelling components behave under the pyramid and prism operations. As a consequence we obtain a concise recursion for the shelling components of the cube.
October 1 Richard Ehrenborg
University of Kentucky
Hamiltonian cycles on Archimedean solids are twisting free
We prove that a Hamiltonian cycle on the faces of an Archimedean solid is twisting free, that is, when returning to the first facet of the cycle, it has the same orientation as in the beginning. We also explore a continuous analogue on the unit sphere.
October 8 Carl Lee
University of Kentucky
Open problems for convex polytopes I'd love to see solved -- slides of talk by Gil Kalai
I will present Gil Kalai's wish list of interesting problems to be conquered in the area of convex polytopes.
October 15 Margaret Readdy
University of Kentucky
Manifold arrangements
We determine the cd-index of the induced subdivision arising from a manifold arrangement. This generalizes earlier results in several directions: (i) One can work with manifolds other than the n-sphere and n-torus, (ii) the induced subdivision is a Whitney stratification, and (iii) the submanifolds in the arrangement are no longer required to be codimension one. This is joint work with Richard Ehrenborg.
October 29 Robert Davis
University of Kentucky
The h*-vector of Gorenstein polytopes with regular, unimodular triangulations
A problem of recent importance has been to identify when a lattice polytope has a symmetric and unimodular h*-vector. In the 1960s Stanley showed that symmetry occurs exactly when the polytope is Gorenstein. In 2003, Athanasiadis gave sufficient conditions for unimodality. In this talk, we will discuss a result by Bruns and Römer that generalizes these conditions.
November 19 Yue Cai
University of Kentucky
A combinatorial approach for q-Stirling numbers
There are many combinatorial interpretations of Stirling numbers of the first and second kinds. In this talk I will focus on the interpretations of de Médicis and Leroux involving 0-1 tableaux and 0-1 matrices. I will describe how they use them to prove identities involving Stirling numbers, such as orthogonality and Carlitz' identity.

Talk at 1 pm -- Note change in time, but not place!

November 27 Brad Fox
University of Kentucky
The cd-index of Bruhat intervals
The cd-index of an Eulerian poset is a polynomial in noncommuting variables that is used to encode data on the number of chains within that poset. I will focus in this talk on a particular class of Eulerian posets involving the Bruhat order of Coxeter groups. I will use poset operations such as zipping and the pyramid operation to develop Reading’s recursive formula for the cd-index of Bruhat intervals.

Talk at 4 pm on Tuesday -- Note change in day & time, but not place!

Seminar Archive

We would like to thank Benjamin Braun (NSA 2013-2015), Richard Ehrenborg (NSF DMS-0902063) and Margaret Readdy (Simons Foundation Collaboration Grant 2011--2016) who continue to generously support our speakers with their grants. The Discrete CATS seminar is also partially supported by the University of Kentucky Department of Mathematics.

If you wish to be added to the seminar list, please e-mail Margaret Readdy at readdy (at)

Last updated April 5, 2013.