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



Spring 2023

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

Jan 16 MLK Day No seminar. Holiday.

Feb 6 Galen Dorpalen-Barry
Ruhr-Universität Bochum
The Poincaré extended ab-index (Zoom) Talk
Motivated by a conjecture of Maglione-Voll from group theory, we introduce and study the Poincaré-extended ab-index. This polynomial generalizes both the ab-index and the Poincaré polynomial. For posets admitting R-labelings, we prove that the coefficients are nonnegative and give a combinatorial description of the coefficients. This proves Maglione-Voll's conjecture as well as a conjecture of the Kühne-Maglione. We also recover, generalize, and unify results from Billera-Ehrenborg-Readdy, Ehrenborg, and Saliola-Thomas.

This is joint work with Joshua Maglione and Christian Stump.

Feb 13 Gábor Hetyei
UNC Charlotte
Brylawski's tensor product formula for Tutte polynomials of colored graphs (Zoom) Talk
The tensor product of a graph and of a pointed graph is obtained by replacing each edge of the first graph with a copy of the second. In his expository talk we will explore a colored generalization of Brylawski's formula for the Tutte polynomial of the tensor product of a graph with a pointed graph and its applications. Using Tutte's original (activity-based) definition of the Tutte polynomial we will provide a simple proof of Brylawski's formula. This can be easily generalized to the colored Tutte polynomials introduced by Bollobás and Riordan. Consequences include formulas for Jones polynomials of (virtual) knots and for invariants of composite networks in which some major links are identical subnetworks in themselves.

All results presented are joint work with Yuanan Diao, some of them are also joint work with Kenneth Hinson. The relevant definitions and the fundamental results used will be carefully explained.

Gábor Hetyei will be a visitor of Richard Ehrenborg and Margaret Readdy in early March.

Feb 20 President's Day -- No Meeting No seminar.

Feb 27 KOI Combinatorics Lectures Local Organizing Committee Meeting

Mar 6 Thomas McConville
Kennesaw State U
Lattices on shuffle words Talk
The shuffle lattice is a partial order on words determined by two common types of genetic mutation: insertion and deletion. Curtis Greene discovered many remarkable enumerative properties of this lattice that are inexplicably connected to Jacobi polynomials. In this talk, I will introduce an alternate poset called the bubble lattice. This poset is obtained from the shuffle lattice by including transpositions. Using the structural relationship between bubbling and shuffling, we provide insight into Greene's enumerative results. This talk is based on joint work with Henri Mülle.

Visitor of Khrystyna Serhiyenko.

Mar 10 Daniel Tamayo
Université Paris-Saclay
On some recent combinatorial properties of permutree congruences of the weak order
(1:00 pm in 745 POT.)
Since the work of Nathan Reading in 2004, the field of lattice quotients of the weak order has received plenty of attention on the combinatorial, algebraic, and geometric fronts. More recently, Viviane Pons and Vincent Pilaud defined permutrees which are combinatorial objects with nice combinatorial properties that describe a special family of lattice congruences. In this talk we will give a brief introduction into the world of (permutree) lattice congruences, how they lead to structures such as the Tamari and boolean lattice, followed by connections to pattern avoidance, automata and some examples of sorting algorithms.

Visitor of Martha Yip.

Mar 10 Yannic Vargas
TU Graz
Hopf algebras, species and free probability
(1:45 pm in 745 POT.)
Free probability theory, introduced by Voiculescu, is a non-commutative probability theory where the classical notion of independence is replaced by a non-commutative analogue ("freeness"). Originally introduced in an operator-algebraic context to solve problems related to von Neumann algebras, several aspects of free probability are combinatorial in nature. For instance, it has been shown by Speicher that the relations between moments and cumulants related to non-commutative independences involve the study of non-crossing partitions. More recently, the work of Ebrahimi-Fard and Patras has provided a way to use the group of characters on a Hopf algebra of "words on words", and its corresponding Lie algebra of infinitesimal characters, to study cumulants corresponding to different types of independences (free, boolean and monotone). In this talk we will give a survey of this last construction, and present an alternative description using the notion of series of a species.

Visitor of Martha Yip.

Mar 13 SPRING BREAK No seminar.

Mar 20 KOI Combinatorics Lectures Local Organizing Committee Meeting

Mar 27

Steven Karp
University of Notre Dame
q-Whittaker functions, finite fields, and Jordan forms
The q-Whittaker symmetric function associated to an integer partition is a q-analogue of the Schur symmetric function. Its coefficients in the monomial basis enumerate partial flags compatible with a nilpotent endomorphism over the finite field of size 1/q. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, which we call the q-Burge correspondence. In the q -> 0 limit, we recover a known description of the classical Burge correspondence (also called column RSK). We use the q-Burge correspondence to prove enumerative formulas for certain modules over the preprojective algebra of a path quiver.

This is joint work with Hugh Thomas.

Mar 31

Mihai Ciucu
Indiana University
Cruciform regions and a conjecture of Di Francesco
The problem of finding formulas for the number of tilings of lattice regions goes back to the early 1900's, when MacMahon proved (in an equivalent form) that the number of lozenge tilings of a hexagon is given by an elegant product formula. In 1992, Elkies, Kuperberg, Larsen and Propp proved that the Aztec diamond (a certain natural region on the square lattice) of order n has 2n(n+1)/2 domino tilings. A large related body of work developed motivated by a multitude of factors, including symmetries, refinements and connections with other combinatorial objects and statistical physics. It was a model from statistical physics that motivated the conjecture which inspired the regions we will discuss in this talk.

A recent conjecture of Di Francesco states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted Tn, are obtained by starting with a square of side-length 2n, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order n-1. Inspired by the regions Tn, we construct a family of cruciform regions Cm,na,b,c,d generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of the region T_n is a divisor of the number of tilings of the cruciform region C2n-1,2n-1n-1,n,n,n-2, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture.

Apr 1 KOI Combinatorics Lectures!

Apr 3 William Dugan
UMass Amherst
William Dugan is a student of Alejandro Morales who is funding this visit.

Apr 4 William Gustafson
University of Kentucky
745 POT, 10 am

Apr 5 Ana Garcia Elsener
Universidad Nacional de Mar del Plata
TBA; Joint with Algebra Seminar
Visitor of Khrystyna Serhiyenko.

Apr 6 Ford McElroy
University of Kentucky
307 POT, 3 pm

Apr 10 Williem Rizer
University of Kentucky
Qualifying Exam, TBA

Apr 17 Marta Pavelka
U Miami
Marta Pavelka is a student of Bruno Benedetti who is funding this visit.

Apr 24

Seminar Archive

Margaret Readdy is organizing the Discrete CATS Seminar in Spring 2023. If you wish to be added to the seminar list or give a talk, please e-mail her at margaret.readdy (at)

The Discrete CATS seminar is partially supported by the University of Kentucky Department of Mathematics.

Last updated March 7, 2023.