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

DISCRETE CATS SEMINAR
WHERE CATS = COMBINATORICS, ALGEBRA, TOPOLOGY & STATISTICS!


745 PATTERSON OFFICE TOWER


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 Talk
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
COLLOQUIUM
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!
Website

Apr 3 William Dugan
UMass Amherst
RESCHEDULED

Apr 4 William Gustafson
University of Kentucky
DOCTORAL DEFENSE
Lattice Minors and Eulerian Posets

(10 am, 745 POT)

We study a partial ordering on pairings called the uncrossing poset, which first appeared in the literature in connection with a certain stratified space of planar electrical networks. We begin by examining some of the combinatorial properties of the uncrossing poset, and then proceed to study the structure of lower intervals. Certain lower intervals are isomorphic to the poset of simple vertex labeled minors of an associated graph. Inspired by this structure we define a notion of minors for lattices enriched with a generating set which abstracts the notion of simple vertex labeled minors of a graph. It is shown that the minor poset of any generator enriched lattice is isomorphic to the face poset of a regular CW sphere, hence minor posets are Eulerian.

We also introduce an operation called weak contractions engineered to commute with deletions. From this we define and study the weak minor poset for generator enriched lattices, many of the results for minor posets have analogues for weak minor posets. Returning to the uncrossing poset we introduce an algebraic object as an abstraction of minor posets of generator enriched lattices which is used to describe to the uncrossing poset and its lower intervals.

Apr 5 Ana Garcia Elsener
Universidad Nacional de Mar del Plata
skew-Brauer graph algebras; Joint with Algebra Seminar
Brauer graph algebras are defined by combinatorial data based on graphs:

Underlying every Brauer graph algebra is a finite graph, the Brauer graph, equipped with with a cyclic orientation of the edges at every vertex and a multiplicity function. This combinatorial data encodes much of the representation theory of Brauer graph algebras and is part of the reason for the ongoing interest in this class of algebras. A known result by Schroll states that Brauer graph algebras, with multiplicity function one, give rise to all possible trivial extensions for gentle algebras. On the other hand, Geiss and de la Pena studied a generalization of gentle algebras called skew-gentle algebras.

In our ongoing project we establish the right definition of skew-Brauer graph algebra in such a way that the result by Schroll can be enunciated in this context. That is, A is a skew-Brauer graph algebra with multiplicity function equal to one if and only if it is the trivial extension of a skew-gentle algebra. Moreover, the family of skew-Brauer graph algebras with arbitrary multiplicity function generalizes the family of Brauer graph algebras with arbitrary multiplicity function.

(Joint work with Victoria Guazzelli from Universidad Nacional de Mar del Plata, Argentina, and Yadira Valdivieso Diaz from Universidad de Puebla, México)

Visitor of Khrystyna Serhiyenko.

Apr 6 Ford McElroy
University of Kentucky
MASTERS EXAM
The Eulerian Transformation and Real-Rootedness
(3 pm in 307 POT)

Many combinatorial polynomials are known to be real-rooted. Many others are conjectured to be real-rooted. The Eulerian Transformation is a map from A:R[t] --> R[t] generated by A(tn)= An(t), the nth Eulerian polynomial. Brenti (1989) conjectured that the Eulerian Transformation preserves real-rootedness. In the 2022 paper The Eulerian Transformation by Brändén and Jochemko, they disprove Brenti's conjecture and make one of their own. In the talk, we will look at
(i) polynomial properties related to real-rootedness,
(ii) Brändén and Jochemko's counterexample to Brenti's conjecture
(iii) evidence Brändén and Jochemko provide to support their conjecture.
Apr 10 Williem Rizer
University of Kentucky
QUALIFYING EXAM
Combinatorics of the Positroidal Stratification of the Totally Nonnegative Grassmannian

The Grassmannian has been an object of much interest in algebra, geometry, and combinatorics. We can decompose the Grassmannian into matroid strata, in which each element of the stratum has the same set of nonzero Pl├╝cker coordinates corresponding to the bases of a matroid. If we restrict to the elements of the Grassmannian with nonngeative coordinates, the corresponding matroids are called positroids. Postnikov revealed a family of combinatorial objects that can be used to parameterize positroidal cells. Recently, various authors have studied objects (X-diagrams and LACD colored permutations) that are in correspondence with this family, though the bijections exhibited do not commute with one another. In this talk we discuss all of these objects, the bijections between them, the information they reveal about positroids, and the possible connections and generalizations we can make to fold these newer objects neatly into the family.

Apr 17 Marta Pavelka
U Miami
2-LC triangulated manifolds are exponentially many
We introduce "t-LC triangulated manifolds" as those triangulations obtainable from a tree of d-simplices by recursively identifying two boundary (d-1)-faces whose intersection has dimension at least d - t - 1. The t-LC notion interpolates between the class of LC manifolds introduced by Durhuus-Jonsson (corresponding to the case t = 1), and the class of all manifolds (case t = d). Benedetti-Ziegler proved that there are at most 2^(N d^2) triangulated 1-LC d-manifolds with N facets. Here we show that there are at most 2^(N/2 d^3) triangulated 2-LC d-manifolds with N facets.

We also introduce "t-constructible complexes", interpolating between constructible complexes (the case t = 1) and all complexes (case t = d). We show that all t-constructible pseudomanifolds are t-LC, and that all t-constructible complexes have (homotopical) depth larger than d - t. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen-Macaulay.

This is joint work with Bruno Benedetti. Details of the proofs and more can be found in our paper of the same title.

Marta Pavelka is a student of Bruno Benedetti who is funding this visit.

Apr 18 Chloé Napier
U Kentucky
MASTERS EXAM
New Interpretations of the Two Higher Stasheff-Tamari Orders
(2:30 pm; 745 POT)

In 1996, Edelman and Reiner defined the two higher Stasheff-Tamari orders on triangulations of cyclic polytopes and conjectured that they are equal. In 2021, Nicholas Williams defined new combinatorial interpretations of these two orders to make the definitions more similar. He builds upon the work by Oppermann and Thomas in the even dimensional case of giving an algebraic analog to these orders using higher Auslander-Reiten Theory. He then gives a completely new result for the odd dimensional case. In this talk, we will discuss the combinatorial interpretations of the even dimensional case and motivate the odd dimensional case and algebraic analog by example.

Apr 24 William Dugan
UMass Amherst
Faces of generalized Pitman-Stanley polytopes
The Pitman-Stanley polytope is a polytope whose integer lattice points biject onto the set of plane partitions of a certain shape with entries in {0 ,1}. In their original paper, Pitman and Stanley further suggest a generalization of their construction depending on $m \in {\mathbb N}$ whose integer lattice points biject onto the set of plane partitions of the same shape having entries in $\{ 0 , 1, ... , m \}$. In this talk, we give further details of this generalized Pitman-Stanley polytope, $PS_n^m(\vec{a})$, demonstrating that it can be realized as the flow polytope of a certain graph. We then use the theory of flow polytopes to describe the faces of these polytopes and produce a recurrence for their f-vectors.

William Dugan is a student of Alejandro Morales who is funding this visit.

May 2 Ben Reese
University of Kentucky
DOCTORAL DEFENSE
Geometry of Pipe Dream Complexes

(2 pm, 745 POT)

In this dissertation we study the geometry of pipe dream complexes with the goal of gaining a deeper understanding of Schubert polynomials. Given a pipe dream complex PD(w) for w a permutation in the symmetric group, its boundary is Whitney stratified by the set of all pipe dream complexes PD(v) where v > w with respect to the strong Bruhat order. For permutations w in the symmetric group on n elements, we introduce the pipe dream complex poset P(n). The dual of this graded poset naturally corresponds to the poset of strata associated to the Whitney stratification of the boundary of the pipe dream complex of the identity element. We next consider pattern avoidance results, including different values of the 132-pattern occurrence and generalizations. We end with a study the facet-ridge diagram of a family of permutations.









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) uky.edu

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


Last updated April 16, 2023.