Date: Thu., Jan. 18, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:Neil Sloane, The OEIS Foundation and Rutgers University-
Title: Eric Angelini's Comma Sequence
Abstract: The Comma Transform of a sequence replaces each comma between the terms by the number formed by concatenating the single digits to the left and right of the comma. (E.g., the Comma Transform of the even numbers is 2, 24, 46, 68, 81, 1, ....) The remarkable "comma sequence" is defined by the property that it starts with 1 and its first differences equal its Comma Transform. If there is a choice, choose the smallest possibility. It contains exactly 2137453 terms! This talk, based on joint work with Eric Angelini, Michael Branic ky, Giovanni Resta, and David W. Wilson, will analyze this and related sequences.
Date: Thu., Feb. 1, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
George Spahn, Rutgers University
Title: Counting Maximal Seat Assignments that obey Social Distancing
Abstract: If I'm proctoring an exam and write on the board that students may not sit adjacent to another student, what should I expect the density of students to be in the seats? We analyze this question by counting maximal arrangements and then show how to generalize the method to answer other related questions. (Joint work with Doron Zeilberger).
Date: Thu., Feb. 15, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Peter Olver, University of Minnesota
Title: Allen Tannenbaum and Computer Vision
Abstract: I will present some of the late Allen Tannenbaum's fundamental contributions to image processing and computer vision, concentrating on our collaborative research. Topics include use of symmetry groups and nonlinear partial differential equations, image denoising and segmentation, and differential invariant signatures for object recognition and symmetry detection, including invariant numerical schemes and later applications to jigsaw puzzles, broken bone assembly, and cancer detection
Date: Thu., Feb. 29, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Jesús Guillera, University of Zaragoza
Title: Bilateral rational Ramanujan series and their p-adic mates
Abstract: We conjecture p-adic identities associated to bilateral rational Ramanujan-like series. Then, we show how to recover the rational Ramanujan series from their p-adic mates
Date: Thu., March 7, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: AJ Bu, Rutgers University
Title: Studying the Area under (Generalized) Dyck Paths
Abstract: I will be presenting my work (along with some joint work with Doron Zeilberger) on how to use symbolic computation to study the area under generalized Dyck paths (i.e. paths in the xy-plane from the origin to (n,0) with an arbitrary set of atomic steps and that never go below the x-axis).
Date: Thu., March 28, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Adam Zsolt Wagner, Worcester Polytechnic Institute.
Title: Reinforcement learning and pattern finding in combinatorics
Abstract: We will look at two ways we can use tools from machine learning to help us with research in combinatorics. First we discuss reinforcement learning, a method that gives us a way to check conjectures for counterexamples efficiently. While it usually does not perform as well as other simpler methods, there have been several examples of projects in the past few years where RL was crucial for success. In the second half of the talk we will consider the following question of Ellenberg: at most how many points can we pick in the N by N grid, without creating an isosceles triangle? The best known constructions, found by computer searches for small values of N, clearly follow a pattern which we do not yet understand. We will discuss how one can train transformers to understand this pattern, and use this t rained transformer to help us find a bit better constructions for various N. This is joint work with Jordan Ellenberg, Marijn Heule, and Geordie Williamson
Date: Thu., April 11, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Natalya Ter-Saakov, Rutgers University
Title: Eric Angelini's Comma Sequence Part 2
Abstract: At the beginning of the semester, Neil Sloane introduced the comma sequence (introduced to him by Eric Angelini) that works as follows. Choose your favorite number in your favorite base. Now we'll build a sequence from it with the following rule: the difference between consecutive terms is equal to the concatenation of the digits on either side of the comma between them. If you chose 1 and base 10, then your next terms would be 12 because 12-1=11 and then 35 because 35-12=23. This sequence will continue for 2,137,452 steps, then fails to find a suitable successor for the value 99,999,945. We will discuss some problems presented by Sloane including a proof that all comma sequences in bases 3 through 19 are finite. Based on joint work with Robert Dougherty-Bliss.
Date: Thu., April 25, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Eugene Zima, Wilfrid Laurier University
Title: Effectively multiplication- and division-free residue number systems
Abstract: Several methods of selection of moduli in modular arithmetic are considered. With the proposed choice of moduli both modular reduction of an integer and reconstruction from modular images are accelerated. Special attention is paid to the moduli of the forms 2n ± 1 and 2n ± 2k ± 1. Different schemes of choice of these types of moduli and algorithms for conversion of arbitrary precision integers into the modular representation and back are considered. Results of experimental implementation of a two-layer modular arithmetic in GMP system are discussed.
Date: Thu., Sept. 12, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Howard Cohl, NIST
Title: Dick Askey (1933-2019) and what I've learned about him and his life
Abstract: Richard (Dick) Askey had monumental influence on the mathematics associated with special functions and orthogonal polynomials. I will describe his legacy.
Date: Thu., Sept. 19, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Frank Calegari, University of Chicago
Title: Searching for sequences: Irrationality beyond Apery
Abstract: In 1978, Apery found a "miraculous" proof that zeta(3) is irrational, by finding an explicit pair of sequences of rational numbers a_n and b_n satisfying a recurrence relation so that their ratio a_n/b_n converged to zeta(3) "too quickly" for zeta(3) to be rational. Given another such pair of sequences, it is easy to verify experimentally whether or not the same "miracle" occurs. The problem is, it seems very hard in practice to find such miraculous sequences whose ratio converges to other interesting Dirichlet L-values. In recent work with Vesselin Dimitrov and Yunqing Wang, we have found (more or less) a weaker condition on the sequences a_n and b_n which implies rationality, and applied this to show that such numbers like
L(2,chi_{-3}) = 1/1^2 - 1/2^2 + 1/4^2 - 1/5^2 + 1/7^2 - 1/8^2 + ,,,
are irrational. The goal of this talk will be to sketch the basic idea, but the main point of the talk will be to explain how to detect sequences which could at least *plausibly* establish irrationality of interesting constants, and also where one might try to look for them.
Date: Thu., Sept. 26, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Rodrigo A. Perez, Indiana University, Indianapolis
Title: A two variable Vandermonde decomposition of q-binomials emerging from a complex dynamics problem
Abstract: When a holomorphic function $f:C \to C$ has a fixed point f(0)=0 with derivative λ=f'(0) of unit size, the question arises of conjugating f to the rotation z ->λ z. This is possible when the argument of λ has good approximation properties; eg, when it is Diophantine. The largest domain of conjugation is known as a Siegel disk.
A famous open problem is to give bounds on the size of Siegel disks. As a concrete case, if $Arg(\lambda)$ is the Golden Ratio, does the Siegel disk contain a disk of radius 1/4?
In the talk I will explore a circle of ideas emerging from our approach to this problem: The value 1/4 is connected to the growth of Catalan numbers enumerating binary trees. A consequence of this correspondence is a 2 variable version of the Vandermonde convolution for (deformed) q-binomials. The work is a joint collaboration with M. Aspenberg, Lund University.
Date: Thu., Oct. 3 2024, 5:00pm (Eastern Time): No talk (rosh hashana)
Date: Thu., Oct. 10, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Neil Sloane, The OEIS Foundation and Rutgers University
Title: A Nasty Surprise in a Sequence, and Other Recent OEIS Stories
Abstract: It was a busy summer. 1. Having raised the necessary endowment, the OEIS is now looking to hire a managing editor. 2. Dampening Down Diverging Series: making the harmonic series and others converge. 3. Covering a sequence (e.g. (n,prime(n))) with straight lines. 4. An innocent-looking definition leads to the Riesel and Sierpinski problems and some astronomical primes. (With thanks to M. Alekseyev, L. Blomberg, L. Brown, D. McCarty, B. McEachen, R. Sigrist, et al.)
Date: Thu., Oct. 17, 2024, 2:00pm (Eastern Time) NOTE SPECIAL TIME
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speakers: See abstract below.
Title: Surprise Party for Dominique Foata on his 90th birthday
Abstract: Originally it was scheduled to be a talk by Doron Zeilberger, but instead it became a surprise party, where Doron Zeilberger, George Andrews, Richad Stanely (via email), Ira Gessel, Mireille Bousquet-Melou, Peter Paule, Jiang Zeng, David Bressoud, Jacques Desarmenien, and Volker Strehl, each gave a four-minute mini-talk/speech followed by a ten minute tribute by Donald Knuth.
Note: This talk was also scheduled in Partion Theory, q-series, and Related Topics organized by William J. Keith.
schedule and linked slides
Zoom surprise party
Date: Thu., Oct. 24, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Amir Alexander, UCLA
Title: The Experimental Geometer: How John Wallis Saved Mathematics for the Royal Society
Abstract: The members of the early Royal Society championed an experimental approach to the study of nature as the proper path to the advancement of knowledge and the preservation of civic peace. Mathematics was admired, but also feared, as dangerously dogmatic and coercive. John Wallis, the leading mathematician in the group, set out to reconcile his field with the ideals of the early Royal Society by developing a radical new approach. Whereas traditional mathematics prided itself on irrefutable deductive proofs, Wallis's approach relied on material intuition, inductive reasoning, and truth-claims founded on consensus, not coercion. It was a new mathematics modeled on the Society's experimental philosophy
Date: Thu., Oct. 31, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Peter Cameron, University of St. Andrews
Title: Finding the jewel in the lotus
Abstract: There has been a lot of recent work about graphs defined on groups so as to reflect some of the group structure; the most famous is the commuting graph, where two group elements are joined if they commute. These graphs typically have huge automorphism groups, but can be tamed by twin reduction. I will describe one case where a beautiful graph is obtained (longer abstract).
Date: Thu., Nov. 7, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Andrew Sills, Georgia Southern University
Title: q-Factorization of power series
Abstract:
In The Theory of Partitions, p. 98, Ex. 2, George Andrews points out that any power series with constant term 1 has a unique factorization in the form
1 + r(1)*q + r(2)*q^2 + r(3)*q^3 + . . . = (1-q)^{-a_1} * (1-q^2)^{-a_2} * (1-q^3)^{-a^3} * . . . .
He then suggests an algorithm to calculate the r(n) given the a_i .
In his qseries.m Maple package, Frank Garvan programmed the inverse algorithm, i.e.
given the r(n), find the a_i. Shashank Kanade and Matthew Russell used this algorithm extensively in their discovery of many new Rogers--Ramanujan type identities, where the a_i form a discrete periodic function with respect to a fixed modulus.
In 1954, G. Meinardus published an asymptotic formula for the r(n) in terms of the a_i.
Recently, I found an exact formula for the r(n) in terms of the a_i and vice versa, which I will
share after presenting some background material.
This work is part of a larger ongoing project joint with Robert Schneider and Hunter Waldron of
Michigan Tech.
lecture
Date: Thu., Nov. 14, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Umesh Shankar, IIT Bombay
Title: Odds and Evens: enumeration results for the alternating subgroup
Abstract: The study of the distribution of statistics over the symmetric group is nearly a century old and has given us a lot of elegant theorems and beautiful bijections. These distributions enjoy a lot of exciting properties such as log-concavity, gamma-positivity, real rootedness etc. In this talk, we will look at a variety of enumeration results for the subset of even permutations and make a case for the study of statistics over them.
Date: Thu., Nov. 21, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Pat Devlin, Swarthmore College
Title: Games Played Randomly-Chomp and Nim
Abstract: In this talk, we discuss combinatorial games where both players move randomly (each turn, independently selecting a legal move uniformly at random), and we will emphasize the "experimental math" approach that was the backbone of our results. We provide closed-form expressions for the expected number of turns in a game of Chomp with any starting condition. We also derive and prove formulas for the win probabilities for any game of Chomp with at most two rows. Additionally, we completely analyze the game of nim under random play by finding the expected number of turns and win probabilities from any starting position. No familiarity with probability is required, as the talk quickly reduces to studying certain recurrence relations that naturally arise from each game.
Joint work with Paulina Trifonova.
Date: Thu., Dec. 5, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Eric Rowland, Hofstra University
Title: Combinatorial structure behind Sinkhorn limits
Abstract: The Sinkhorn limit of a positive square matrix is obtained by scaling the rows so each row sum is 1, then scaling the columns so each column sum is 1, then scaling the rows again, then the columns again, and so on. It has been used for almost 90 years in applications ranging from predicting telephone traffic to machine learning. But until recently, nothing was known about the exact values of its entries. In 2020, Nathanson determined the Sinkhorn limit of a 2 x 2 matrix, and Ekhad and Zeilberger determined the Sinkhorn limit of a symmetric 3 x 3 matrix. We were able to determine the Sinkhorn limit of a general 3 x 3 matrix, and the result suggests the general form for n x n matrices. In particular, the coefficients reflect new combinatorial structure on sets of minor specifications. This is joint work with Jason Wu; Here is our paper .
Date: Thu., Dec. 12, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Éric Fusy, Laboratoire d'Informatique Gaspard Monge, Marne-la-Vallée
Title: Enumeration of corner polyhedra
Abstract: will present results on the exact and asymptotic enumeration of corner polyhedra, a special class of simple orthogonal polyhedra introduced by Eppstein and Mumford, whose enumeration can be considered as a topological analogue of counting plane partitions.
Joint work with Erkan Narmanli and Gilles Schaeffer
Date: Thu., Dec. 19, 2024, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Neil Sloane, The OEIS Foundation and Rutgers University
Title: Eric Angelini's Greatest Sequences
Abstract: Over a period of 20 years, Eric Angelini contributed 1700 brilliant, clever, witty sequences to the OEIS. I will discuss seven of them: "Which terms are primes?", The Jungfrau, Solar Flares, Even Digit Next Bigger, Look Left, Delete Repeated Digits, The Rigidity of the Okapi.