Date: Thu., Jan. 23, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Stoyan Dimitrov, Rutgers University
Title: Picking, posing and attacking natural problems in discrete mathematics: from insightful bijections to black-box help from machine learning
Abstract: We will walk through several combinatorial results. Half of them have important motivation coming from computer science and the other half explain surprising observations made by experimentation. We will begin by a high-level discussion on how one shall pick or pose his problems, and end by sharing more about an exciting machine learning technique that may change the way we approach combinatorial (and mathematical) problems in general. Some exciting open questions will also be mentioned.
Date: Thu., Jan. 30, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title: A map of the holonomic forest - searching for irrationality with Conservative Matrix Fields
Abstract: Is this number irrational?
This question has been troubling mathematicians since the time of Pythagoras, and inspired a search for good
rational approximations of constants. Examining the literature on irrationality, we notice a recurring theme: holonomic sequences.
These recursive sequences have dominated this field, generating very good approximations. In this talk we shall detail an experimental journey into the
"forest" of holonomic functions. In our exploration, we will pick up a powerful object - the Conservative Matrix Field. With this map in hand,
we shall re-discover many known Diophantine approximations (Apery, Beukers, Zudilin, Aptekareve), and some new ones as well.
If time permits, we will detail more deeply the conjectured properties of these Matrix Fields.
Speaker: Shachar Weinbaum, Technion (member of the Ramanujan Machine group)
Date: Thu., Feb. 6, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
David Broadhurst, Open University, UK
Title: Resurgent integer sequences
Abstract: In combinatorics, we happily manipulate formal power series, taking no heed of whether they might converge. Applied mathematicians encounter series with no radius of convergence, about which they worry. Jean Ecalle mediates between these communities, by telling us about resurgent trans-series. I shall give an account of how an integer sequence from a problem in physics exhibits resurgence.
Title: A mirror step variant of gambler's ruin
Abstract: Consider a gambler who starts with x dollars. At each gamble, the gambler either wins a dollar with probability 1/2 or loses a dollar with probability 1/2. The gambler's goal is to reach N dollars without first running out of money (i.e., hitting 0 dollars). If the gambler reaches N dollars, we say that they are a winner. The gambler continues to play until they either run out of money or win. This scenario is known as the gambler's ruin problem, first posed by Pascal. We consider a new generalization of the gambler's ruin problem. A particle starts at some point x on a line of length N. At each step, the particle either moves from x to x-1 with probability q1, or moves from x to x+1 with probability q2, or moves from x to N-x with probability p where 0
Title: Ascent sequences avoiding a set of length-3 patterns
Abstract: See here
Title: Young Tableau Reconstruction Via Minors
Abstract:
Starting point of our talk will be a short discussion of the classical 15 puzzle. It turns out that similar problems about sliding numbered tiles around in a box are of enormous relevance to combinatorial algebra in the context of Young tableaux. In particular, we are going\
to present "The Tableau Reconstruction Problem," posed by Monks (2009). Starting with a standard Young tableau T of size n, i.e., an arrangement of n numbered tiles, a 1-minor of T is a tableau obtained by first deleting any tile of T, and then sliding the remaining tiles \
into position and renumbering them according to some rules. This can be iterated to arrive at the set of k-minors of T. The problem is this: given k, what are the values of n such that every tableau of size n can be reconstructed from its set of k-minors? For k=1, the prob\
lem was recently solved by Cain and Lehtonen. In this talk, we discuss the problem for k=2, proving the sharp lower bound n ≥ 8. In the case of multisets of k-minors, we also give a lower bound for arbitrary k, as a first step toward a sharp bound in the general multis\
et case
Title: The Mathematics of Adriano Garsia (1928-2024)
Abstract: Adriano Garsia (wiki) started out
in Harmonic Analysis, but soon enough saw the light and became one of the leaders of Enumerative and Algebraic Combinatorics. In this tribute,
students, collaborators, and colleagues, will try and describe some of his seminal contrubutions. After the end of the formal part,
there would be plenty of time for an "open stage" where everyone (including his 36 PhD students) would have the opportunity to
say a few words in Adriano's memory.
Title: A central limit theorem in the framework of the Thompson group F
Abstract: The classical central limit theorem states that the average of an infinite sequence of independent and identically distributed random variables, when suitably rescaled, tends to a normal distribution. In fact, this classical result can be stated purely alg\
ebraically, using the combinatorics of pair partitions. In the 1980s, Voiculescu proved an analog of the central limit theorem in free probability theory, wherein the normal distribution is replaced by Wigner's semicircle distribution. Later, Speicher provided an
algebraic proof of the free central limit theorem, based on the combinatorics of non-crossing pair partitions. Since then,
various algebraic central limit theorems have been studied in noncommutative probability, for instance, in the context of symmetric groups. My talk will discus\
s a central limit theorem for the Thompson group F, and show that the central limit law of a naturally defined sequence in the group algebra of F is the normal distribution. Our combinatorial approach employs abstract reduction systems to arrive at this result.
Title: Combinatorial exploration and permutation classes
Abstract:
Permutations, words, set partitions, and other such families of objects often play a role in diverse subfields of mathematics, physics and computer science. When the structure of the object under investigation is known there are well-established tools, such as symbolic and \
analytic combinatorics, that derive an enumeration, asymptotics, and the ability to randomly generate instances of the objects. However, the initial step from a definition of the object to a structural description is often ad-hoc, human-staring-at-a-blackboard type of work.\
This is the gap combinatorial exploration attempts to fill.
Combinatorial exploration is a domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and provide an open-source Python impleme\
ntation. As a prerequisite, we build up a new theoretical foundation for combinatorial decomposition strategies and combinatorial specifications.
Combinatorial exploration has been most extensively applied to permutation classes, rederiving hundreds of results in the literature as well as discovering many novel results (which can be found on permpal.com). As well as unifying earlier methods, one key advantage of our \
approach is its ability to utilise a growing library of strategies in a simultaneous manner to build a greater understanding of the structure of the permutation classes.
Date: Thu.,April 3, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title: Some positivity conjectures for symmetric functions motivated by
classical theorems from the analytic theory of polynomials
Abstract:
I recall some classical theorems from the analytic theory of polynomials,
and then ask whether they can be "upgraded" from pointwise positivity
to coefficientwise positivity. This leads to a series of positivity
conjectures for symmetric functions. This is joint work with Yusra Naqvi.
Title: Some surprises in lattice problems
Abstract: I present some (idealized) problems from condensed matter physics and cluster chemistry that lead to mathematical insights that seem to fly in the face of the scientists' intuition
Date: Thu.,April 24, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title: Modular arithmetic with trinomial moduli
Abstract:
A popular approach to speed up computations is to reduce the input modulo several relatively prime numbers, do arithmetic on the residues, then reconstruct the result at the end. This improves the running time if the moduli (and their pairwise inverses) have "nice" binary s\
hapes, but only a limited number of such moduli are known. I will discuss a new system of moduli related to a family of trinomials. This system has shown promising real-world performance, can produce arbitrarily many moduli, and its elements can be computed quickly. I will \
mention some theoretical limitations of the system based on roots of unity and graph colorings, and point out what remains to be discovered.
Joint work with Mits Kobayashi, Natalya Ter-Saakov, and Eugene Zima.
Pablo Blanco's talk information:
Title: Generating functions of sequences relating to spanning trees in certain graph families
Abstract: Generating functions of sequences relating to spanning trees in certain graph families
Abstract: Kirchhoff's Matrix Tree Theorem allows us to compute the number of spanning trees of a graph by looking at its Laplacian matrix.
For certain graph families (in our case, powers of cycles and paths), which are represented by finitely many states, we know by the Transfer Matrix Method
that a rational generating function exists for sequences arising from structures in the family. Such a generating function can be found by computing sufficiently many
terms of the sequence. In joint work with Doron Zeilberger, we found generating functions for the number of spanning trees and for a leaf-parameter by experimental methods.
Aurora Hively's talk information:
Title: Experimenting with Permutation Wordle
Abstract: Consider a game of permutation wordle in which a player attempts to guess a secret permutation in Sn in as few guesses as possible.
In each round, the guessing player is told which indices of their guessed permutation are correct. How can we optimize the player's strategy? Samuel Kutin and Lawren Smithline propose a strategy
called cyclic shift in which all incorrect entries are shifted one index to the right in successive guesses, and they conjecture its optimality. We investigate this conjecture by formalizing what a
"strategy" looks like, performing experimental analysis on inductively constructed strategies, and taking advantage of Kutin-Smithline's findings related to Eulerian numbers.
Date: Thu., May 8, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title: The mathematics and physics of Joel Lebowitz
Abstract: Joel Lebowitz (b. May 10, 1930) has made many important contributions to both mathematics and physics, some of them will be outlined in this talk.
Date: Thu., Sept. 11, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title: Cutting a Pancake Using an Exotic Knife
Abstract:
In the first chapter of their classic book "Concrete Mathematics", Graham, Knuth, and Potashnik consider the maximum number of pieces that can be obtained from a pancake by
making n cuts with a knife blade that is straight, or bent into a V, or bent twice into a Z. We extend this work by considering knives, or "cookie-cutters", of even more exotic shapes, including a k-armed V,
a chain of k connected line segments, or are shaped like one of the upper-case letters A, E, L, M, T, W, or X, or like a regular polygon, a circle, figure-8,
pentagram, or hexagram. In most cases a counting argument combined with Euler's formula produces an explicit formula for the maximum number of pieces.
Title: A heuristic link between divisor counts and prime densities in sequences
Abstract:
I introduce a heuristic principle I call "probabilistic continuation" and conjecture a striking asymptotic equivalence: the density of primes in a well-behaved integer sequence appears to match a structural ratio derived from the divisor counts of its terms.
The appeal of this conjecture is practical. Determining prime densities usually demands heavy analytic machinery (as in the Prime Number Theorem),
whereas the associated divisor ratio is far easier to evaluate. If the conjecture is correct, this ratio could thus provide a simpler proxy for fundamental density measures.
I will present the main conjecture, show its consistency with classical results (PNT, PNT in arithmetic progressions, prime distribution in quadratic-residue sequences), and discuss its coherence with
Hardy-Littlewood-type conjectures.
Date: Thu., Sept. 25, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title:A journey with MacMahon and Ramanujan series
Abstract:
MacMahon introduced a generalized notion of divisor sums
that are easily interpreted in terms of integer partitions. In this talk,
we take the audience on a tour that includes our encounters with a variety
of subjects, such as "prtition Eisenstein series", symmetric functions theory,
and topological genera. We highlight the advantages of being an experimentalist, a humble disciple of Zeilberger.
This presentation spans joint work with several co-authors.
Title: Okey and Random Combinatorial Games
Abstract: Okey is a Turkish tile-based game where players aim to empty their hand by forming sets of same numbered tiles, or runs of same colored tiles.
I'll first showcase my experimentations with generalized Okey, and then move onto some experimenting with random combinatorial games. we will delve into statistics about game durations and winners, along with the evolution of random combinatorial games
Title: Identity Found by Proving Identities
Abstract: At the 3rd Formal Power Series and Algebraic Combinatorics conference,
that took place in Bordeaux in 1991, Doron Zeilberger gave an invited talk with the title
"Identities in Search of Identity". At the same time, his seminal paper
on the holonomic systems approach to special function identities was
published and, together with Herbert Wilf, he developed the WZ theory
for proving hypergeometric summation identities. During the following 35
years, this theory has been considerably extended and refined, and
evolved into its own research area within symbolic computation. We
recapitulate its evolution, highlight its main achievements, and discuss
some recent trends. We then turn our attention to applications in
combinatorics, with a special emphasis on the treatment of determinants
and Pfaffians, which became amenable to symbolic methods via the
holonomic ansatz: the sought identity may be transformed into a set of
summation identities, which themselves can be proven algorithmically.
This procedure is elucidated with prominent examples, such as the
q-enumeration of totally-symmetric plane partitions, the counting of
configurations in the twenty-vertex model, and the evaluation of
binomial determinants emerging from rhombus tilings.
Title: Computer algebra tools for Feynman Integrals
Abstract: Physicists have been using computers extensively both for simulations
and numerical calculations. I will describe the on-going effort of harnessing the
power of symbolic computation to further our knowledge of particle physics.
[Joint work beteen RISC(Linz) and DESY (Deutsches Elektronen-Synchrotron, J. Bluemlein and P. Marquard)]
i>Date: Thu., Oct. 30, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title: Powers of 2 in Balanced Grid Colourings
Abstract: See here and
here .
slides
Title: Cutting Rectangles into Two Congruent Pieces
Abstract: In the March 2025 issue of Pour La Science (the French analog of Scientific American), Jean-Paul Delahaye, (the French (and contemporary) analog of Martin Gardner),
solved (in collaboration with his wife, Martine Raison), the problem of counting the number of ways of
cutting a 3 by 2n checkerboard into two (connected) congruent pieces, and proved the simple explicit
formula 2n+1-n-1. When we asked Delahaye whether he knew the answer for a four by n rectangle, he replied that he has no clue, but the problem
seems to him to be très difficile. We will describe how, with the right grammar, and some help from our silicon friends, we solved this
challenging problem.
Date: Thu., Nov. 13, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title:Maximizing the number of integer pairs summing to powers of 2 via graph labeling and solving restricted systems of linear (in)equations
Abstract: We address the problem of finding sets of integers of a given size with a maximum number of pairs summing to powers of 2. By fixing particular pairs, this problem reduces to finding a labeling of the vertices of a given graph with pairwise distinct integers\
such that the endpoint labels for each edge sum up to a power of 2. We propose an efficient algorithm for this problem, which at its core relies on another algorithm that, given two sets of linear homogeneous polynomials with integer coefficients, computes all variable ass\
ignments to powers of 2 that nullify polynomials from the first set but not from the second. With the proposed algorithms, we determine the maximum size of graphs of order n that admit such a labeling for all n<=21, and construct the maximum admissible graphs for n<=20. We \
also identify the minimal forbidden subgraphs of order n<=11, whose presence prevents the graphs from having such a labeling.
Date: Thu., Nov. 20, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title: (Non)commutative integrable systems and Catalan numbers
Abstract: will discuss connections of Catalan numbers with solutions
of some systems of
differential equations over (non)commutive algebras.
Joint work with Ilia Gaiur and Vladimir Rubtsov
Date: Thu., Dec. 4, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Title: Continuous parking sequences
Abstract:We introduce the notion of continuous parking sequences as a continuous analogue of parking functions. We allow the cars to have different lengths and the street to be longer
than the sum of the lengths of the cars. We determine the volume of the set of continuous parking sequences. Furthermore, we study the distribution where the first car, respectively, the last car would like to park. Joint work with Richard Ehrenborg and Stephan Wagner.
Title: Guessing with little data
Abstract: Automated guessing is a very popular tool in experimental mathematics. Given the first few terms of an infinite sequence, it tries to find a recurrence that the sequence is
in some way likely to satisfy. If the sequence under consideration indeed satisfies a recurrence and a sufficient number of initial terms is supplied as input, then the method is bound to find the correct answer.
But sufficiently many initial terms are not always available. Together with Christoph Koutschan, we have proposed a variant of the classical method that needs fewer terms
and catches some of these cases. We will explain the idea behind this variant and present some conjectures that have been found with the method.
Date: Thu., Feb. 20, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Toufik Mansour, University of Haifa
Date: Thu., Feb. 27, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Daniel Herden, Baylor University
i>Date: Thu., March 6, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
coordinator and MC: Angela Hicks. Here is the program
(that includes the list of speakers)
Date: Thu., March 13, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Arundhathi Krishnan,Mary Immaculate College, Thurles, Ireland
Date: Thu., March 27, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Christian Bean, Keele University
Speaker: Alan Sokal, University College London.
Date: Thu.,April 17, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Michael Kiessling, Rutgers University
Speaker: Robert Dougherty-Bliss, Dartmouth College
Date: Thu., May 1, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speakers:
Pablo Blanco (5:00-5:23 pm) and Aurora Hively (5:25-5:48 pm) (both from Rutgers University)
Speaker:
Sheldon Goldstein, Rutgers University
Fall 2025 Semester
Speaker: Neil Sloane, The OEIS Foundation and Rutgers University.
This is based on joint work with David O. H. Cutler. If time permits I will also discuss A386482, an exciting new version of the EKG sequence.
Date: Thu., Sept. 18, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker:
Benoit Cloitre (independent researcher)
Speaker: Tewodros Amdeberhan, Tulane University
Date: Thu., Oct. 9, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Nuray Kutlu [linkedin], Amazon
Date: Thu., Oct. 16, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Christoph Koutschan, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria
Date: Thu., Oct. 23, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Carsten Schneider, RISC-Linz (Austria)
Speaker: Nikolai Beluhov, Cambridge University
Date: Thu., Nov. 6, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speakers: Robert Dogherty-Bliss, Dartmouth College, and Natalya Ter-Saakov, Rutgers University.
(Joint work by the two of us with Doron Zeilberger)
Speaker: Max Alekseyev, George Washington University
Speaker: Vladimir Retakh, Rutgers University
Speaker: Mei Yin, University of Denver
Date: Thu., Dec. 11, 2025, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]
Speaker: Manuel Kauers, Johannes Kepler University, Linz, Austria