*Date:* Jan. 27, 2022, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Donald E. Knuth, Stanford University

*Title*: Tchoukaillon numbers

*Abstract*: Mancala games have fascinated people worldwide for centuries,
and Tchoukaillon is a particularly nice specimen of such a game.
I will indicate how it might help to answer the following basic
question about which nothing is currently known:
Are there bipartite matching problems for which the Hopcroft--Karp
algorithm actually has nonlinear running time?

*Title*: Determinant solutions for nonlinear differential equations in
the commutative and noncommutative case

*Abstract*: James Joseph Sylvester was the first one to find a determinant
solution of a system of nonlinear differential equations which is now
known as Toda system. I will talk about commutative and noncommutative
generalizations of his result. I will also discuss connections between
Toda systems, Painlevé equations and orthogonal polynomials

*Title*: Pattern avoidance in parking functions

*Abstract*: We extend the classical definition of patterns in permutations to parking functions. In particular we study parking functions that avoid
permutations of length 3. A number of well-known combinatorial sequences arise in our analysis, and this talk will highlight several enumeration results that were conjectured
and/or proved collaboratively with the computer. This project is joint work with Ayomikun Adeniran.

*Title*: Some trigonometric identities associated with the roots of unity

*Abstract*: Consider the complete graph whose vertices are the n-th roots of unity in the
complex plane. To every edge between a pair of vertices, associate a weight that is a
given even non-zero power of the length of the edge. Sum the weights over all pairs of
vertices (i.e., over all edges). The result, determined in all generality by Johann Brauchart
around 2014 (special cases were known a century before) is always a finite expression in
integer powers of n --- thanks to the trivial zeros of Riemann's zeta function. This talk is
intended to serve an exciting appetizer to the audience, who hopefully will wish to go for
the full meal by reading Johann's papers. Some MAPLE experiments do feature in this talk.

*Title*: The method of brackets. How to integrate in an easy manner

*Abstract*: This talk will discuss a (relatively) new method for integration. It was developed by
Ivan Gonzalez as part of his PhD thesis in the analysis of Feynman diagram. A large collection
of examples will be given to illustrate its power.

*Title*: Sorting probabilities for Young diagrams

*Abstract*: Sorting probability for a partially ordered set P is defined as the min |Pr[x < y] - Pr[y < x]|
going over all pairs of elements x,y in P, where Pr[x < y] is the probability that in an uniformly random linear extension (extension to total order) x appears before y.
The celebrated 1/3-2/3 conjecture states that for every poset the sorting probability is at most 1/3, i.e. there are two elements x and y, such that 1/3 ≤ Pr[x < y] ≤ 2/3.
The asymptotic extension of this conjecture states that the sorting probability goes to 0 as the width (maximal antichain) of the poset grows to infinity.
We will prove the last conjecture for Young diagrams, where the linear extensions are Standard Young Tableaux. We also discuss notable special cases relating to random walks.
Based on joint works with Swee Hong Chan and Igor Pak.

*Date:* March 10, 2022, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Jay Pantone, Marquette University

*Title*: Combinatorial Exploration: An Algorithmic Framework for Enumeration

*Abstract*: Combinatorial structures are ubiquitous throughout mathematics. Graphs, permutations, words, and other such
families of combinatorial objects often play a central role in work from many different fields. The study of enumerative combinatorics is concerned
with the elucidation of structural properties of these families, including counting, classification, and limiting behavior.

Combinatorial Exploration is a framework that unifies the often ad-hoc methods used in enumerative combinatorics. In this talk we will explain how Combinatorial Exploration works, how it can be automated, and how it is applied to the study of pattern-avoiding permutations to prove new results and reprove dozens of old ones. We will also discuss the new web database PermPAL that catalogs these results

*Date:* March 24, 2022, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Natalya Ter-Saakov, Rutgers University

*Title*: Origami flip graphs of flat-foldable vertices

*Abstract*: In the study of flat origami, each crease pattern has an associated set of valid mountain-valley assignments - ones that will allow it to fold flat.
We study how these assignments for single-vertex crease patterns are related to one another through face flips, where flipping a face means switching the assignment of all
bordering creases. Specifically, we explore the origami flip graph OFG(C) of a given crease pattern C where each vertex is a valid mountain-valley assignment for C
and two vertices are adjacent if their assignments differ by a single face flip. We show how different origami flip graphs of single-vertex crease patterns are related and provide an edge count for the maximal case.
Joint work with Thomas C. Hull, Manuel Morales, and Sarah Nash

*Date:* March 31, 2022, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* Richard Ehrenborg, University of Kentucky

*Title*: Sharing Pizza in n Dimensions

*Abstract*:
We introduce and prove the n-dimensional Pizza Theorem.
This is joint work with Sophie Morel and Margaret Readdy.
longer abstract

*Title*: Minimal Circuits for Boolean Functions of Few Variables

*Abstract*: Using enumerative techniques, we produce minimum-size circuits for
every Boolean function of four or fewer variables and every monotone Boolean
function of five variables. This talk will focus on the modeling decisions and
optimization techniques used to produce the catalogs of circuits.

*Title*: The Meta C-Finite Ansatz

*Abstract*: The Fibonacci numbers F(n) satisfy the famous recurrence F(n + 2) = F(n + 1) + F(n). The "C-finite ansatz" tells us that the family of sequences F(2n), F(3n), F(4n), ..., along with their sums and products satisfy similar recurrences. However, even more is true. We will show that the recurrences satisfied by F(n*i) and F(n*i) * F(n*j), for any C-finite sequence F, satisfy meta recurrences which lead to generating function and summation identities.

*Title*: Exact Specral Gap and Gap Eigenfunctions for a Class of Graphs
Generalizing the Johnson Graphs

*Abstract*: We consider a class of graphs describing a system of N particles with r
prescribed energy levels that interact through pair "collisions"
in which they exchange energies. In the mathemtical physics problem
associated to this model, the spectral gap of the graph Laplacian
on the connected components is of interest. We present a simple method
for determining the exact spectral gaps in all cases,
and all of the gap eigenfunctions. The method also yields some
additional information on the spectrum. We discuss some
conjectures supported by Maple computations. This is joint work with
Michael Loss.

Guest Lecture by Dr. Neil Sloane in Dr. Z.'s Grad class (Thurs. , April 28, 2022)

*Speaker:*
Karl-Dieter Crisman, Gordon College

*Title*: Voting on Cyclic Orders, Representations, and Ties

*Abstract*: In social choice theory there has been recent interest in voting on various combinatorial objects. One such object is cyclic orders, which may be thought of (informally) as ways of seating people around a table.

In this talk we will introduce a bit of algebraic voting theory, which profitably uses representation theory to classify (linear) voting systems. However, linear algebra can't solve every problem by itself, so we also discuss some of the patterns computer exploration discovered in investigating all possible complete ties under a particular system.

Joint work with Abraham Holleran, Micah Martin, and Josephine Noonan

*Title*: Game Theory: Foundation and Applications

*Abstract*: Game Theory is a set of mathematical tools for analyzing interactive decision making. The flexibility of this methodology has
yielded a range of cross disciplinary applications, from social science to systems and computer science. In this talk, I give a general survey of game theoretic modeling and its applications.

*Title*: New Sequence Problems and Solutions from 2022

*Abstract*: I'll discuss new results on the Stepping Stones problem, the amazing Magic Carpet problem from combinatorial geometry,
a simple number-theory sequence which is crying out to be analyzed, the solution to a 60-year old problem of Kaprekar, and several versions of the Lexicographically
Earliest Sequence question, one of which (the Binary Two-Up Sequence) has just been solved