# Discrete Math Seminar (DMS)

The **Discrete Math Seminar (DMS)** is intended for Kennesaw State faculty working in the various areas of algebra, number
theory, and discrete mathematics to get together to discuss their current work or
related questions. Seminars often involve advanced mathematical knowledge. However,
the seminars are open to anyone who is interested in attending.

**Upcoming Events**

#### Wednesday, November 16, 2022

**SPEAKER: Adam Gregory**, University of Florida**TIME/LOCATION:**2:30-3:20pm in D-117 Mathematics building, Marietta campus and**online in Microsoft Teams****TITLE:***"Lenart’s Bijection via Bumpless Pipe Dreams"***ABSTRACT:**Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semi-standard tableaux via maps due, respectively, to Lenart and Weigandt. Recently, Gao and Huang have defined a bijection between the former two sets. In this talk we will show that the Gao-Huang bijection preserves the associated tableaux for vexillary permutations, giving a new proof of Lenart’s result. This is joint work with Zachary Hamaker.

## Past Events

#### Friday, September 4, 2020

**SPEAKER: Dr. Andrew Wilson**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"The combinatorics of harmonic polynomials"***ABSTRACT:**Polynomials in n variables carry an action of the symmetric group on n symbols. If we take the orthogonal complement of the polynomials which are invariant under this action, we get what is called the space of harmonic (or co-invariant) polynomials. In 1942, Emil Artin showed that this harmonic space has dimension n!. Recent work has extended this result to allow the symmetric group to act "diagonally" on multiple sets of variables and to allow some of these variables to anti-commute, motivated by ideas from mathematical physics. I will give an overview of this work, focusing on the combinatorial objects which arise, including permutations, parking functions, and ordered set partitions.

#### Friday, October 2, 2020

**SPEAKER: Dr. Mikhail Lavrov**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Monochromatic squares, Hales-Jewett numbers, and why 5 is harder than 4"***ABSTRACT:**I'll begin this talk by talking about a Ramsey problem in the hypercube. This is a distant cousin of the problem for which the record-setting Graham's number was invented, but we'll be less ambitious and accordingly our upper bound will be much smaller: 78. (This result is joint work with John Mackey and Mitchell Lee.)

The proof technique generalizes to solve a different problem. The Hales-Jewett theorem is a Ramsey result for monochromatic lines in a high-dimensional grid; it implies many other results in Ramsey theory, but is very hard to find good upper bounds for. We show how to bound the (two-color) Hales-Jewett number when the grid has side length 4. Finally, we'll see why generalizing to side length 5 or more is hard and needs new ideas.

#### Friday, November 6, 2020

**SPEAKER: Dr. Julianne Vega**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Integer Decomposition Property of Schur and Symmetric Grothendieck Polynomials"***ABSTRACT:**In this presentation we will consider Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we show that these polytopes have integer decomposition property. This work is joint with Margaret Bayer, Bennet Goeckner, Su Ji Hong, Tyrrell McAllister, McCabe Olson, Casey Pickney, and Martha Yip.

#### Friday, February 5, 2021

**SPEAKER: Dr. Mikhail Lavrov**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Comparisons, edge coloring, and the Erdos-Szekeres theorem"***ABSTRACT: Link to pdf abstract**

#### Friday, February 19, 2021

**SPEAKER: Dr. Erik Westlund**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Extendability of k-suitable matchings in Cartesian products"***ABSTRACT: Link to pdf abstract**

#### Friday, April 2, 2021

**SPEAKER: Dr. Thomas McConville**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:**"Determinantal formulas with major indices"**ABSTRACT: Link to pdf abstract**

#### Friday, April 16, 2021

**SPEAKER: Dr. Richard Ehrenborg**, University of Kentucky**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Sharing pizza in n dimensions"***ABSTRACT: Link to pdf abstract**

#### Friday, September 3, 2021

**SPEAKER: Dr. Thomas McConville**, Kennesaw State University**TIME/LOCATION:**3:00-4:00pm in D-120 Mathematics building, Marietta campus**TITLE:***"Enumerative aspects of bubbling and shuffling"***ABSTRACT:**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 incorporating a third type of mutation, namely transposition. 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.

#### Friday, September 24, 2021

**SPEAKER:****Dr. Andrés Vindas Meléndez**, University of California, Berkeley and MSRI**TIME/LOCATION:**3:00-4:00pm in D-120 Mathematics building, Marietta campus and streaming online in Microsoft Teams**TITLE:***"Ehrhart Theory of Paving Matroid Polytopes"***ABSTRACT:**Ehrhart theory is a topic in geometric combinatorics which involves the enumeration of lattice points in integral dilates of polytopes. We introduce pan-handle matroids, which can be understood as a certain lattice-path matroid. We provide a formula for the Ehrhart polynomial of the matroid polytope of pan-handle matroids. This Ehrhart polynomial allows us to obtain a formula for matroid polytopes of paving matroids. No prior knowledge will be assumed and the talk will aim to be accessible to non-experts. This is joint work with Mohsen Aliabadi, Derek Hanely, Jeremy L. Martin, Daniel McGinnis, Dane Miyata, George D. Nasr, and Mei Yin.

#### Friday, October 15, 2021

**SPEAKER:**Dr. Zhu Cao, Kennesaw State University**TIME/LOCATION:**3:00-4:00pm in D-120**Mathematics building, Marietta campus**and online in Microsoft Teams**TITLE:***"Exact covering systems, quadratic forms, and identities for the Rogers-Ramanujan functions"***ABSTRACT:**The Rogers-Ramanujan functions satisfy the famous Rogers-Ramanujan identities as well as Ramanujan's continued fraction. These identities have combinatorial interpretations and combinatorial proofs. As an example of applications, Rogers-Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics. Ramanujan established a list of forty identities involving the Rogers-Ramanujan functions. In the past, people have developed various tools to attack the forty identities.

From the point of view of exact covering systems (ECS), we can extend the definition of the equivalence of quadratic forms. Furthermore, we show that there are natural connections among ECS, quadratic forms, and product identities for theta functions. In the past several years, I have unified most of the previous approaches and provided nice explanations for most of the identities among the forty identities. The applications of our main result are not just restricted to identities for the Rogers-Ramanujan functions. We can also work on n-ary quadratic forms (especially ternary quadratic forms) and present a list of new identities.

#### Friday, October 29, 2021

**SPEAKER:****Dr. Anthony Bonato**, Ryerson University**TIME/LOCATION:**3:00-4:00pm online in Microsoft Teams**TITLE:***"Pursuit-evasion games on graphs"***ABSTRACT:**In pursuit-evasion games, a set of pursuers attempts to locate, eliminate, or contain an evader in a network. The rules, specified from the outset, greatly determine the difficulty of the questions posed above. For example, the evader may be visible, but the pursuers may have limited movement speed, only moving to nearby vertices adjacent to them.

Central to pursuit-evasion games is the idea of optimizing certain parameters, whether they are the search number, burning number, or localization number, for example. We report on progress in several pursuit-evasion games on graphs and conjectures arising from their analysis. Finding the values, bounds, and algorithms to compute these graph parameters leads to topics intersecting graph theory, the probabilistic method, and geometry.

#### Friday, November 12, 2021

**SPEAKER: Dr. Paul Pollack**, University of Georgia**TIME/LOCATION:**3:00-4:00pm online in Microsoft Teams**TITLE:***"Multiplicative orders mod p: what not everyone knows"***ABSTRACT:**If*a*is an integer coprime to*m*, the multiplicative order of*a*mod*m*is the least positive integer*k*with*a^k==1*mod*m*. I will discuss some lesser-known problems and results about multiplicative orders modulo prime numbers*p*. Particular attention will be paid to the following kind of question: For two fixed integers*a*and*b*, how do the multiplicative orders of*a*mod*p*and*b*mod*p*compare, as*p*varies? This represents joint work with Matt Just and Sergei Konyagin.

#### Friday, February 4, 2022

**SPEAKER: Dr. Ernie Croot**, Georgia Institute of Technology**TIME/LOCATION:**3:00-4:00pm in D-107**Mathematics building, Marietta campus**and online in Microsoft Teams**TITLE:***"On a problem of Graham on the p-divisibility of central binomial coefficients ${2n \choose n}$"***ABSTRACT: Link to pdf abstract**

#### Friday, February 18, 2022

**SPEAKER:**Dr. Cvetelina Hill, Kennesaw State University**TIME/LOCATION:**3:00-4:00pm in D-107**Mathematics building, Marietta campus**and online in Microsoft Teams**TITLE:***"Tropical convex hull of polyhedral sets"***ABSTRACT:**In this talk we will discuss the interaction between tropical and classical convexity, with a focus on the tropical convex hull of convex sets and polyhedral complexes. We will give a vertex description of the tropical convex hull of a line segment and a ray, show that for sets in two dimensions tropical convex hull and ordinary convex hull commute, and characterize tropically convex sets in any dimension. We will give a combinatorial description for the dimension of a tropically convex fan, and the dimension of the tropical convex hull of an ordinary affine space. Finally, we introduce a tropical proof for a lower bound on the degree of a fan tropical curve.

#### Thursday, March 3, 2022

**SPEAKER: Dr. Emily Heath**, Iowa State University**TIME/LOCATION:**3:30-4:30pm in D-116**Mathematics building, Marietta campus**and online in Microsoft Teams**TITLE:***"The Erdős-Gyárfás Problem on Generalized Ramsey Numbers"***ABSTRACT:**A*(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 the function*f(n, p, q)*, first introduced by Erdős and Shelah, which is the minimum number of colors needed for a*(p, q)*-coloring of the complete graph*K_n*. In 1997, Erdős and Gyárfás initiated the systematic study of this function. Among other results, they gave upper and lower bounds on*f(n, p, p)*, which are still the best known bounds for general*p*today. In this talk, I will give an overview of this problem and describe recent improvements on the probabilistic upper bound of Erdős and Gyárfás for several small cases of*p*. This is joint work with Alex Cameron.

#### Friday, March 4, 2022

**SPEAKER: Felix Clemen**, University of Illinois**TIME/LOCATION:**3:00-4:00pm in D-107**Mathematics building, Marietta campus**and online in Microsoft Teams**TITLE:***"Triangles in Planar Point Sets"***ABSTRACT:**A triangle*T*is*epsilon*-similar to another triangle*T'*if their angles pairwise differ by at most*epsilon*. Given a triangle*T*,*epsilon>0*and a natural number*n*, Bárány and Füredi asked to determine the maximum number of triangles being*epsilon*-similar to*T*in a planar point set of size*n*. We determine this quantity for almost all triangles*T*and sufficiently small*epsilon*. Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof. This is joint work with József Balogh and Bernard Lidický.

#### Friday, April 8, 2022

**SPEAKER:**Dr. Michael Joseph, Dalton State College**TIME/LOCATION:**3:00-4:00pm in D-107**Mathematics building, Marietta campus**and online in Microsoft Teams**TITLE:***"Dynamics in Posets: Promotion, Rowmotion, Evacuation, and Rowvacuation"***ABSTRACT:**Dynamical algebraic combinatorics is an area focused on group actions on (usually) finite sets. There are several phenomena that one encounters surprisingly often, especially homomesy (where a statistic has the same average across every orbit), dyclic sieving, and periodicity. Promotion and rowmotion are two cyclic group actions on the order ideals of a finite poset which have received considerable attention over the last decade. There are also the related involutions of rowvacuation and evacuation, which have received far less attention until recently. Rowmotion and rowvacuation together generate a dihedral group action on the order ideals of a graded poset, and promotion and evacuation do similarly for rowed-and-columned posets. In fact, the names promotion and evacuation come from Schützenberger's related actions on linear extensions, but we will focus on order ideals instead. This talk will be an introduction to the many patterns and mysteries in dynamical algebraic combinatorics.

#### Friday, April 15, 2022

**SPEAKER: Christina Giannitsi**, Georgia Institute of Technology**TIME/LOCATION:**3:00-4:00pm online in Microsoft Teams**TITLE:***"Improving and Maximal Inequalities for Primes in Progressions"***ABSTRACT: Link to pdf abstract**

#### Wednesday, September 14, 2022

**SPEAKER: Dr. Thomas McConville**, Kennesaw State University**TIME/LOCATION:**2:30-3:20pm in D-117 Mathematics building, Marietta campus and online in Microsoft Teams**TITLE:***"The fundamental theorem of finite semidistributive lattices"***ABSTRACT:**A finite lattice has two binary operations, meet and join, which may or may not satisfy a distributive law. Birkhoff neatly characterized lattices for which the distributive law is always satisfied. More recently, many interesting families of lattices have been constructed that are semidistributive, which implies that the distributive law only holds sometimes. Motivated by some examples of these lattices in Representation Theory, Reading, Speyer, and Thomas discovered a remarkable characterization of finite semidistributive lattices. In this talk, I will present their theorem and some applications.

#### Wednesday, October 5, 2022

**SPEAKER: Dr. Zachary Hamaker**, University of Florida**TIME/LOCATION:**2:30-3:20pm in D-117 Mathematics building, Marietta campus and online in Microsoft Teams**TITLE:***"On characters of permutation statistics"***ABSTRACT:**Functions of permutations are studied in a wide variety of fields including probability, statistics and theoretical computer science. I will introduce a method for studying such functions using representation theory and symmetric functions. As a consequence, one can extract detailed information about asymptotic behavior of many permutation statistics with respect to non-uniform measures that are invariant under conjugation. The key new tool is a combinatorial formula called the path Murnaghan-Nakayama rule that gives the Schur expansion of a novel basis of the ring of symmetric functions. No background about representation theory or symmetric function theory will be assumed. This is joint work with Brendon Rhoades.

#### Wednesday, October 12, 2022

**SPEAKER: Dr. Khrystyna Serhiyenko**, University of Kentucky**TIME/LOCATION:**2:30-3:20pm in D-117 Mathematics building, Marietta campus and online in Microsoft Teams**TITLE:***"Triangulations of flow polytopes, ample framings, and gentle algebras"***ABSTRACT:**The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. We characterize the DAGs that admit ample framings, and we enumerate the number of ample framings for a fixed DAG. Moreover, we establish a connection between maximal simplices in DKK triangulations and tau-tilting modules for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that their flow polytopes are Gorenstein and have unimodal Ehrhart h^{∗}-polynomials. This is joint work with Matias von Bell, Benjamin Braun, Kaitlin Bruegge, Derek Hanely, Zachery Peterson, and Martha Yip.

#### Wednesday, October 26, 2022

**SPEAKER: Ashley Tharp**, North Carolina State University**TIME/LOCATION:**2:30-3:20pm in D-117 Mathematics building, Marietta campus and online in Microsoft Teams**TITLE:***"Arcs and shards"***ABSTRACT:**The group of permutations is the canonical example of a finite Coxeter group, and each permutation can be represented visually by a noncrossing arc diagram. Each diagram encodes the canonical join representation of its permutation, and diagrams can be used to understand lattice congruences on the weak order of type A, equivalence relations that respect the meet and join. This talk will provide an overview of existing results on noncrossing arc diagrams for permutations and present two new constructions of noncrossing arc diagrams which relate a given pair of diagrams. We state dual theorems about the relevance of these constructions to the shard intersection order of type A, a partial order weaker (given by fewer relations) than the weak order. We also present analogs of noncrossing arc diagrams (joint with Barnard and Reading) for signed permutations (type B) and even-signed permutations (type D). These results lead to work in progress exploring sublattice relationships between shard intersection orders on different finite Coxeter groups.