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.
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
Title: Parameter estimation in discrete-time systems
Abstract:The parameter identifiability problem for a dynamical system is to determine whether the parameters of the system can be found from a given subset of variables of the system. Verifying whether the parameters are indentifiable is a necessary first step before trying to estimate the values of the parameters. We will discuss these problems in the context of discrete-time systems with exponential functions.
Title: Sorting probability for Young diagrams
Abstract: Can you always find two elements x,y of a partially ordered set, such that, the probability that x is ordered before y when the poset is ordered randomly, is between 1/3 and 2/3? This is the celebrated 1/3-2/3 Conjecture, which has been called "one of the most intriguing problems in the combinatorial theory of posets". We will explore this conjecture for posets that arise from (skew-shaped) Young diagrams, where total orderings of these posets correspond to standard Young tableaux. We will show that these probabilities are arbitrarily close to 1/2, by using random walk estimates and the state-of-the-art hook-length formulas of Naruse. This is a joint work with Igor Pak and Greta Panova. This talk is aimed at a general audience.
Title: Enumerative combinatorics and coding theory
Subtitle: Eliahou Theory and application to enumerating Legendre Pairs
Abstract: In his seminal 1994 paper entitled "Enumerative combinatorics and coding theory", Shalom Eliahou proposes a theory/method to enumerate the values of a polynomial in n variables with non-negative integer coefficients, assumed on {-1,+1}^n. The method builds an associated binary linear code whose enumeration of certain weight suffices to answer the question. We shall present Eliahou's theory, as well as its application to the problem of enumerating Legendre pairs. For background in Legendre pairs, please consult the ICECA 2022 talk .
Here is a promotional video .
Title: The Arithmetic-Periodicity of the game CUT
Abstract: CUT is a class of partition games played on a finite number of finite piles of tokens. We will discuss some interesting properties of these games.
Date: Nov. 17, 2022, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias 40!/(20!*21!), alias 6564120420 ]
Speaker:
Yaakov Malinovsky, University of Maryland, Baltimore County
Title: Hitting a Prime in 2.43 Dice Rolls (on average) and
on Round-Robin Tournaments with a Unique Maximum Score
Abstract: I will discuss two different topics. The first describing joint work with Noga Alon and the second with John W. Moon. For a detailed abstract see here.
slides for Part I slides for Part II
Title: Unification of Set Partitions
Abstract: In this talk we will consider a unified generating functions for 9 different kinds of set partitions including cyclically ordered set partitions. Such generating function depends on 4 parameters. We consider some properties of this function and provide combinatorial explanation for polynomials generated by this function.
Date: Dec. 15, 2022, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Richard Evan Schwartz, Brown University.
Title: Continued Fractions and the 4-Color Theorem
Abstract: Starting from one of the well-known reformulations of the 4-color theorem, I'll show some experiments I did concerning proper 4-colorings of the sphere triangulations in which the maximum number of triangles around a vertex is 6. I haven't made much progress on understanding all the solutions but I will show off a nice infinite family that gives a novel geometric interpretation of continued fractions.