Date: Thu., Jan. 29, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Drew Sills, Georgia Southern University
Title: Mathematically inspired musical scales
Abstract: We review some basics of "musical physics" from Pythagoras to modern times, and
then examine variations on these ideas whereby we can build nonstandard musical scales inspired by the elementary functions of mathematics. We then show how to build an infinite family of musical scales for which Wendy Carlos's alpha, beta, and gamma scales are special ca\
ses. Some of this work is ongoing and joint with
Robert Schneider, frontman for the indie rock band The Apples in Stereo, and Assistant Professor of Mathematics at Michigan Technological University.
slides
lecture
Date: Thu., Feb. 5, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Matthew C. Russell
Title: Mechanical proofs of partition identities through atomic relations
Abstract: Integer partition and q-series sum-to-product identities, such as the Rogers-Ramanujan identities, lie at the intersection of combinatorics, number theory, and the representation theory of affine Lie algebras. This talk will focus on the method of atomic relations, which has been developed by Shashank Kanade and the author in recent years as a way to prove these identities. In some cases, computer output from this method can guide human-constructed proofs, while in other cases, our proofs are totally computer-generated. Applications of this method to problems involving cylindric partitions and colored partitions will also be discussed.
Special Guest Lecture in Dr. Z.'s experimental math class, Thurs. Feb. 12, 2026, 12:10-1:10 pm
Speaker: Wadim Zudilin, Radboud University, Nijmegen, Netherlands
Title: Jesus Guillera (1955-2026)
slides
[ The people in the picture in slide 16 are, from left to right
1) Jesús Guillera Goyanes 2) Wadim Zudilin 3) Eva A. Gallardo Gutiérrez 4) Francisco Ruiz Blasco 5) Jesús Bastero Eleizalde 6) Javier Cilleruelo (1961-2017) 7) Juan Carlos Peral Alonso or Jose Luis Cuadra
8) Fernando Chamizo Lorente 9) Herbert Wilf
(Thanks to Guruzeta Guillera-Arroita) ]
Title: Counting Colored Tilings on Grids and Graphs
Abstract: In this talk we study a counting problem that originated on Mathematics Stack Exchange: How many ways can a rectangular grid be partitioned into a prescribed number of connected polyominoes when the pieces are colored, and any two pieces that share an edge must have different colors? We organize these numbers using bivariate generating functions, where one variable records the length of the grid and the other records the number of pieces. Using generating functions, we obtain explicit rational expressions in the first nontrivial cases of two and three r\ ows. We then recast the model in graph-theoretic terms by replacing grids with Cartesian products of a fixed graph and a path, and by counting properly colored partitions into connected blocks. This leads to analogous generating functions on graphs, including closed forms f\ or specific families (such as complete graphs), and to a computational framework for exploring further examples. This is joint work with Diego Villamizar (Xavier University of Louisiana
Date: Thu., Feb. 19, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Timothy Chow, Center for Communication Research, Princeton, NJ
Title: Game theory, Scrabble, and poisons
Abstract: We describe two unexpected applications of game theory to recreational mathematics. The first, which is joint work with Scrabble expert Nick Ballard, is a Scrabble position in which the best strategy is mixed (i.e., randomized). Although it is not theoretic\ ally surprising that such positions should exist, ours is the first provably correct explicit example, and even elite Scrabble players find it striking. The second concerns an old but little-known lateral-thinking "poison puzzle" by Michael Rabin, which turns out to have se\ veral alternative solutions to the intended solution. We are led to consider a two-player game with three distinct Nash equilibria, which is difficult to analyze theoretically and begs for experimental investigation
Speaker: Lucy Martinez, Rutgers University
Title: How many coin tosses would you need until you get n Heads or m Tails?
Abstract: How many coin tosses would it take until reaching for the first time either n Heads or m Tails?. Although this setup is related to the classical Problem of points, prior work by Fermat and Pascal focused on the probability of getting n Heads vs getting m Tails, rather than on the duration of the game. Using probabilistic techniques, togeth\ er with symbolic computation, we derive probability generating functions, closed-form expressions, and asymptotic formulas for moments of the stopping time to reach either n Heads or m Tails. In the symmetric case when n=m, we show that the expected duration with a fair coi\ n is a polynomial in p whose coefficients are Catalan numbers. Time permitting, we will also discuss the related problem of obtaining n Heads and m Tails. Joint work with Svante Janson and Doron Zeilberger
Date: Thu., March 5, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Nayda Farnsworth , Colgate University
Title: A Computational Approach to Improving Bounds on the Hales-Jewett Numbers
Abstract: We use SAT solvers to improve bounds of the celebrated Hales-Jewett Numbers, one of the most important numbers in Ramsey Theory.
Date: Thu., March 12, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Nathan Fox, Canisius University
Title: Counting Colored Trees
Abstract: A plane tree is a rooted tree where each node's children have a left-to-right order. Classically, the number of plane trees with n+1 vertices is equal to the nth Catalan number. We can generalize this basic enumeration problem to plane trees with colored vertices. We consider coloring rules that, given the color of the parent node, restrict the choices of how to color the children. This general framework is fertile ground for combinatorial exploration. For one thing, it generalizes many different examples that have been studied in the literature. It also leads to many new results, including bijections with other known problems. In this talk, we will explore various families of coloring rules and explore the integer sequences that enumerate plane trees colored according to those rules. This is joint work with Stoyan Dimitrov, Kimberly Hadaway, Ashley Tharp, and Stephan Wagner.
Date: Thu., March 19, 2026, 5:00pm (Eastern Time)
NO TALK (SPRING BREAK)
Date: Thu., March 26, 2026, 5:00pm (Eastern Time)
NO TALK (because of Lewis lecture)
Date: Thu., April 2, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Victor Miller, Anduril Industries
Title: Summing a challenging series
Abstract: On the math-fun list, Neil Sloane posed the following problem: Let V(n) denote the integer formed by using the base 10 digits of n in base 11. It is classical that the series sum_n 1/V(n) converges. It is challenging to calculate a good approximation to its value. As a second, related, problem find a good approximation to the subseries sum_p 1/V(p), where the sum is over primes. It turned out that the first problem was efficiently solved by two related methods described by Robert Baillie and Jean-Francois Burnol. However, they do not appear to apply to the second sum, since they both depend, implicitly, on the fact that the language of digits in the first problem is a regular language, and a recent result of Thomas Dubbe shows, in a technical sense, that the digits of primes are poorly approximated by a regular language. In this talk I'll describe attempts at approximating the value of the second sum. They involve fractals, Fourier series, the prime zeta function, and the Karamata inequality. The process of analyzing this was helped, considerably, by experimentation, and seeing structure in graphs of quantities related to the series.
Date: Thu., April 9, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Eugene Zima, Wilfrid Laurier University and SCG (UWaterloo)
Title: Sparse and scalable Residue Number Systems from polynomial point of view
Abstract: Residue number systems (RNS) based on pairwise relatively prime moduli are a powerful tool for accelerating integer computations via the Chinese Remainder Theorem. We study families of sparse moduli exhibiting scalable modular inverses that enable the accel\ eration of all aspects of modular arithmetic: conversion to RNS, intra-RNS operations, and reconstruction from modular images.
Our unified approach is to represent moduli as the evaluations of sparse polynomials at powers of 2. Two moduli can be checked for scalability by evaluating a single polynomial resultant. If the polynomials are suitable, one can generate sets of moduli of arbitrary length w\ ith closed-form modular inverses. We also discuss different strategies to evaluate scalable moduli to further eliminate RNS overhead.
We show that this approach is universal by using it to test well-known sets of moduli for scalability.
This is joint work with Robert Dougherty-Bliss and Natalya Ter-Saakov.
slides of main talk slides of mini-talk in honor of Sergei Abramov
Date: Thu., April 16, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Jay Pantone,Marquette University
Title: Computational and Experimental Methods in Permutation Patterns
Abstract: For most of its existence, a hallmark of permutation patterns research has been the use of computers. Our research is regularly made possible by the ability to write a simple script to generate permutations with some certain property, helping us to discover an interesting theorem; or to open up Sage, or Maple, or Mathematica and perform some large generating function calculation; or to use one of the several existing large permutation patterns software libraries to test some intriguing conjectures.
The quest to understand permutation classes has led to the import of computational methods from other areas into the field of permutation patterns, as well as the development of a number of new techniques. Some of these methods produce rigorous results, assuming the correctn ess of the software implementation. Others are experimental in the sense that their output should be considered conjectural. The popularity of permutation patterns has even led to some of these computational techniques making the jump to other areas of combinatorics. This ta lk will survey a collection of these methods, including some developed by myself and my collaborators.
Date: Thu., April 23, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Alex Kontorovich, Rutgers University
Title: Use of Computers in Mathematical Research
Abstract: We'll discuss new uses of technology, including Lean and AI, to aid the research mathematician
Date: Thu., April 30, 2026, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Paul Levrie, KU Leuven, Belgium
Title: On the mathematical legacy of Guillera
Abstract: In this seminar, we will provide a brief introduction to the early history of series with sums of 1/π and 1/π2, we meet Ramanujan, and discuss the role played by the recently deceased Jesus Guillera in this story. We derive closed-form expressions for values of generalized hypergeometric functions that elegantly combine Ramanujan's series and Guillera's series, as well as others that combine Guillera's series with the series with sum 1/π4. For this we use Zeilberger's algorithm. A side result of the method used is a new (?) series for the constant G/π3, that Jesus Guillera probably would have liked, G being Catalan's constant.
[Joint work with John Campbell]