Great Plains Combinatorics Conference 2022

Invited Talks

Jeremiah Bartz, University of North Dakota

Title: Balancing Numbers: Finding Balance in Sequences

Abstract: A common endeavor in mathematics is to identify when two quantities are equal. In 1999, Behera and Panda applied this concept to a certain integer sequence to define the notion of balancing numbers. More precisely, we say the positive integer B is a balancing number if there is a positive integer m such that T(B-1)+T(B)=T(m) where T(k) = k(k+1)/2 is the kth triangular number. For example, 6 is a balancing number since T(5)+T(6)=T(8) or equivalently 1+2+3+4+5=7+8. In this talk, we survey results of balancing numbers and their generalizations. In particular, we identify characteristics that balancing numbers share with Pythagorean triples and the Fibonacci sequence as well as current research directions.

Slides for Jeremiah Bartz's talk are here

Emily Heath, Iowa State University

Title: The Erdős-Gyárfás problem on generalized Ramsey numbers

Abstract: (p, q)-coloring of a graph G is an edge-coloring of G (not necessarily proper) in which each p-clique contains edges of at least q distinct colors. We are interested in finding the minimum number of colors, f(n, p, q), needed for a (p, q)-coloring of the complete graph on n vertices. Erdős and Gyárfás initiated the systematic study of this function in 1997. In this talk, I will survey known results for this Ramsey problem and describe strategies for improving the upper and lower bounds on f(n, p, q).

This is joint work with Alex Cameron and with József Balogh, Sean English, and Robert Krueger.

Slides for Emily Heath's talk are here.

Chris Fraser, Michigan State University

Title: Webs and canonical bases in degree two

Abstract: An ongoing problem in algebraic combinatorics is the construction of "good bases" for representations of simple Lie groups. I will give some background on some approaches to this problem, including an approach via web diagrams. Then I will explain a new result which fits into this framework: Lusztig's canonical basis for the degree two part of the Grassmannian coordinate ring consists of web diagrams.

Slides for Chris Fraser's talk are here.

Jeremy Martin, University of Kansas

Title: Simplicial and Cellular Trees

Abstract: One of the first results in algebraic graph theory is Kirchhoff's Matrix-Tree Theorem, which enumerates spanning trees of a graph in terms of its Laplacian matrix. A classic application is a quick proof of "Cayley's formula" $n^{n-2}$ for the number of labeled trees on $n$ vertices. What about spanning trees of higher-dimensional objects, like

simplicial and cell complexes? It turns out that the same methods work (with some topological wrinkles). I will give an overview of the subject, and if time permits will discuss my recent work with Duval, Kook, and Lee on higher-dimensional electrical potential theory and applications to spanning tree enumeration.

Slides for Jeremy Martin's talk are here.

Anna Weigandt, Massachusetts Institute of Technology

Title: The Castelnuovo-Mumford Regularity of Matrix Schubert Varieties

Abstract: The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchgot, Ren, Robichaux, and St. Dizier, we noted that the regularity of Matrix Schubert Varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for the degrees of symmetric Grothendieck polynomials. In this talk, I will present a combinatorial degree formula for arbitrary Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer.

Slides for Anna Weigandt's talk are here.

Nathan Williams, University of Texas at Dallas

Title: Pop-Tsack Torsing

Abstract: Given a finite irreducible Coxeter group W, we use the W-noncrossing partition lattice to define a Bessis dual version of C. Defant’s notion of a Coxeter pop-stack sorting operator. We show that if W is coincidental or of type D, then the identity element of W is the unique periodic point of this operator and the maximum size of a forward orbit is the Coxeter number of W. In each of these types, we obtain a natural lift from W to the dual braid monoid of W. This is joint work with C. Defant.

Mei Yin, University of Denver

Title: Parking functions: Interdisciplinary connections

Abstract: The topic of parking functions has wide applications in probability, combinatorics, group theory, and computer science. Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Through a combinatorial construction termed a parking function multi-shuffle, we investigate various probabilistic properties of a uniform parking function. Time permitting, algebraic properties of parking functions will also be discussed. Partially based on joint work with Richard Kenyon.

Worksheet for Mei Yin's talk is here.

Poster Session Abstracts

Joseph Bernstein North Dakota State University

Title: P-strict Promotion and B-bounded Rowmotion

Abstract: We define P-strict labelings for a finite poset P as a generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with a toggle action on B-bounded Q-partitions of an associated poset Q. In many nice cases, this toggle action is conjugate to rowmotion. We apply this result to flagged tableaux and Gelfand-Tsetlin patterns, and also show P-strict promotion can be equivalently defined using Bender-Knuth and jeu de taquin perspectives. Joint work with Jessica Striker and Corey Vorland.

Mark Denker, University of Kansas

Title: A Hopf Monoid on Posets with Ordinal Sum

Abstract: A combinatorial Hopf monoid is an algebraic object that keeps track of how combinatorial objects can be combined and broken. We present a new Hopf monoid on posets using the ordinal sum and induced subposet operations, compute its antipode, and demonstrate its relationships with other known Hopf monoids, including the Hopf monoid on graphs and linear orders.

John Forsman, North Dakota State University

Title: Root closure of ideals

Abstract: Given an ideal, its integral closure can be recovered through a variety of processes. In particular, it appears as the degree one graded component in the integral closure of the Rees algebra. By studying the lesser-known root closure ring operation, we find an analogous closure on ideals. We introduce this root closure of an ideal which appears as the degree one graded component of the root closure of the Rees algebra.

Kimberly Hadaway, Iowa State University

Title: An Introduction to Parking Functions

Abstract: In 1966, Alan G. Konheim and Benjamin Weiss defined "parking functions" as follows: We have a one-way, one-lane street with n parking spaces, numbered in consecutive order from 1 to n, and we have n cars in line waiting to park. Each driver has a favorite (not necessarily distinct) parking spot, which we call its preference. We order these preferences in a preference vector. As each car parks, it drives to its preferred spot. If that spot is open, the car parks there; if not, it parks in the next available spot. If a preference vector allows all cars to park, we call it a parking function. In 1974, Henry O. Pollak proved the total number of parking functions of length n, meaning there are n parking spots and n cars, to be (n+1)^(n-1). In this presentation, we describe a recursive formula, expound Pollak's succinct six-sentence proof of an explicit formula, and conclude with a discussion of other parking function generalizations.

J. Martin, Iowa State University

Title: Universal Partial Cycles on De Bruijn Graphs.

Abstract: A universal partial cycle for Aⁿ is a cyclic sequence that covers each word of length n over the alphabet A exactly once, like a De Bruijn cycle except that we also allow a wildcard symbol ◇ that simultaneously represents every letter of A. It is well known that a De Bruijn cycle for {0,1,...,a-1}^n can be represented as a Hamiltonian cycle on the De Bruijn graph B(a,n) or as an Eulerian tour of B(a,n-1). I give analogous representations of universal partial cycles on De Bruijn graphs, including a nondeterministic Euler tour on B(a,n-1) and a concentric circles diagram consisting of a subgraph of B(a,n). I also visit our alphabet multiplier result, showing that, given any universal partial cycle for {0,1,...,a-1}^n, one can construct a universal partial cycle for {0,1,...,ak-1}^n for any natural number k, so each example generates an infinite family of universal partial cycles.

Pratyush Mishra, North Dakota State University

Title: Girth Alternative for HNN extensions

Abstract: The girth of a finite graph is the length of the shortest cycle in it. Graphs with a small degree and large girth are interesting objects; intuitively, these graphs tend to be close to expanders and the latter is an important notion in computer science and combinatorics. In a similar spirit, the girth of a finitely generated group is defined as the supremum of the girths of Cayley graphs where the supremum is taken over all finite sets of generators of the group. This notion of girth for the finitely generated group was first introduced by S. Schleimer in 2003. Later, a substantial amount of work on the girth of finitely generated groups was done by A. Akhmedov, where he introduced the so-called Girth Alternative and proved it for certain classes of groups, e.g. hyperbolic, linear, one-relator, PL_+(I) etc. Girth Alternative is like the well-known Tits Alternative in spirit; therefore, it is natural to study it for classes of groups for which Tits Alternative has been investigated. We will explore the girth of HNN extensions of finitely generated groups in its broadest sense by considering cases where the underlying subgroups are either full or proper subgroups. We will present a sub-class for which Girth Alternative holds. We will also produce counterexamples to show, that beyond our class, the alternative fails in general. This is a joint work with Azer Akhmedov.

Elizabeth Sprangel, Iowa State University

Title: Graph Universal Cycles

Abstract: Graph universal cycles are a graph analogue of universal cycles introduced in 2010. We demonstrate connections between universal cycles and graph universal cycles for threshold graphs and permutation graphs. Additionally, we introduce graph universal partial cycles, a more compact representation of graph classes, which use “do not know” edges. We then show how to construct graph universal partial cycles for labeled and unlabeled graphs, threshold graphs, and permutation graphs.

Kaelyn Willingham, University of Minnesota

Title: Topological Properties of Random Cubical Complexes

Abstract: This poster details the study of topological properties of cubical complexes constructed by a random sampling of square faces from an n-dimensional cube. We start by analyzing the Betti numbers of a given complex as well as the Euler characteristic, in an effort to understand how the homology of a random complex is influenced by the underlying probability distribution by which it was constructed. We are particularly interested in the low-order homology groups H_0, H_1, H_2 which provide meaningful information about the connectivity, cycle, and cavity structure of a cubical complex. In so doing, we derive formulas for the expected mean & variance of the number of isolated k-dimensional components of a random complex under some probability distribution. We conclude by briefly describing future plans to study the possible existence & behavior of nontrivial torsion in homology of a random complex, as well as the Poincare polynomial of a random complex.

Sylvester Zhang, University of Minnesota

Title: Cluster Structures from Decorated super-Teichmüller Spaces

Abstract: Penner’s λ-length coordinates in decorated Teichmüller spaces are known to be cluster algebras. In this poster, I will describe a supersymmetric analogue of cluster algebra structure in decorated super-Teichmüller spaces of Penner and Zeitlin. In particular, we give two combinatorial formulas for super λ-lengths from a triangulated surface: one in terms of super T-path, and the other using double dimers on snake graphs. This is joint work with Nick Ovenhouse and Gregg Musiker.