This page contains links to the slides of the on-line conference
Combinatorics and Algebras from A to Z, July 26-29, 2021, in honor (one year late) of Amitai Regev's 80th birthday and Doron Zeilberger's 70th birthday.
Title: Opening Remarks
Date: Monday, July 26, 2021, 10:10-10:40 EDT,
Speaker: Gil Kalai, The Hebrew University of Jerusalem
Title: My Wonderful Experience with Enumeration, Computers, Algebra, Z, and A
Abstract: I will talk about personal (amateurish at times) experience on how adding weights can help us enumerate, how computers can explore and prove theorems, and how algebra and combinatorics interlace.
Date: Monday, July 26, 2021, 10:40-11:10 EDT
Speaker: Aaron Robertson, Colgate University
Title: The Thread of Ramsey Theory in Zeilberger's Work
Abstract: Zeilberger gave me a Ramsey theory problem as part of my dissertation. I always wondered why, since he did not do Ramsey theory, and all of his other students at around that time were working in the area of permutation patterns. Follow me as I attempt to discover from where this may have evolved.
Date: Monday, July 26, 2021, 11:20-11:50 EDT,
Speaker: Toufik Mansour, University of Haifa
Title: Restricted Permutations, Conjecture of Lin and Kim, and Work of Andrews and Chern
Abstract: In each of the following cases, we will see how computer programming assists with finding the solution. 1. Let wk be the number of distinct Wilf classes of subsets of exactly k patterns in S4 We show that w3= 242, and give the exact values up to w24=1.
2. We compute the distribution of the first letter statistic on nine avoidance classes of permutations, corresponding to two pairs of patterns of length 4. In particular, we show that the distribution is the same for all classes, and is given by a new Schröder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim.
3. We exend the work of Andrews and Chern.
[For a longer version of the abstract, click on "Titles and Anbstracts" here]
Title: Growth Rates of Grids and Merges of Permutation Classes
Abstract: The first general result on growth rates of permutation classes is a consequence of Regev's foundational 1981 work on the asymptotic enumeration of Young diagrams, and gives the growth rate of the class of permutations avoiding any monotone pattern. I will describe a generalization of a theorem of Bevan, which shows that the growth rate of a grid class of permutations is given by the square of the largest singular value of a related matrix. I will then show how this result can be used to rederive several consequences of Regev's results, as well as establish several other results of Bona and others. Joint work with Michael Albert (University of Otago) and Jay Pantone (Marquette University)
Title: Superalgebras Described by Root Data
Abstract: Similarly to the classical theory of semisimple (or Kac-Moody) Lie algebras, we describe a procedure to assign a Lie superalgebra to certain root data. Our notion of root data takes into account the existence of "odd reflections", which are important in the theory of Lie superalgebras. This approach produces so-called "root algebras" such that Kac-Moody (super)algebras are the "minimal ones". The talk is based on a joint project with V. Hinich and V. Serganova.
Date: Tuesday, July 27, 2021, 10:40-11:10 EDT,
Speaker: Eli Aljadeff, Technion
Title: PI Theory and G-graded Division Algebras
Abstract: If G is a finite group and D is a finite dimensional G-graded division algebra over k, then restricting scalars to F=k, , the algebraic closure, we obtain a finite dimensional G-graded simple algebra. In this lecture we address the problem in the opposite direction, namely if A is a finite dimensional G-graded simple algebra over F (with char(F)=0), then under which conditions does it admit a G-graded division algebra form (in the sense of descent theory)? The main tools come from G-graded PI theory. These allow us to construct the corresponding generic objects.Joint work with Karasik.
Date: Tuesday, July 27, 2021, 11:20-11:50 EDT,
Speaker: Miriam Cohen, Ben-Gurion University
Title: Conjugacy Classes and Representations of Hopf Algebras and their Quantum Double
Abstract: [Click on "Titles and Abstracts" in the conference main page]
Date: Tuesday, July 27, 2021, 12:00-12:30 EDT,
Speaker: Allan Berele, DePaul University
Title: Specialized Symmetric Polynomials
Abstract: The problem of determining the relations between symmetric functions in a finite number of not necessarily distinct variables is related to the problem of determining trace identities for diagonal matrices, with a non-standard trace function. The talk will be elementary, with a number of open problems.
Date: Wed., July 28, 2021, 10:00-10:30 EDT,
Speaker: Manuel Kauers, Johaness Kepler University, Linz
Title: Quadrant Walks Starting Outside the Quadrant
Abstract: Throughout his long academic life, Doron has always been very good at thinking ``outside the box''. I am going to present a study of a combinatorial problem where we literally also were thinking ``outside the box'': we consider lattice walks restricted to a quadrant whose starting point is outside the quadrant. This model leads to an interesting functional equation that at first glance looks like it should have an algebraic solution. With the help of the celebrated creative telescoping technology pioneered by Doron, we could show that the solution is in fact transcendental. For certain variations of the problem, we have semi-rigorous proofs that the generating functions are also transcendental, and in one case, we suspect that the generating function is not even D-finite. This is joint work with Manfred Buchacher and Amelie Trotignon, to be published at FPSAC 2021.
Title: Log-convexity P-recursive sequences
Abstract: Based on the asymptotic analysis, we present a method for proving the log-convexity or the log-concavity of P-recursive sequences.
Honoring Doron (prepared by Yuval Roichman, thanks are also due to Gil Zeilberger for providing some of the pictures)
Honoring Amitai (prepared by Yuval Roichman. Thanks to Iris Regev (for the Bach recital), and Eli Aljadeff and Yuri Bahturin for providing some of the pictures)
Date: Thursday, July 29, 2021, 10:00-10:30 EDT,
Speaker: Christoph Koutschan, RICAM (Austrian Academy of Sciences)
Title: Binomial Determinants for Tiling Problems Yield to the Holonomic Ansatz
Abstract: We study some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. By adapting Zeilberger's holonomic ansatz to make it work for these problems, we can take full advantage of computer algebra tools for symbolic summation. As a result, we are able to resolve all open conjectures related to these determinants, including one from 2005 due to Lascoux and Krattenthaler.
Date: Thursday, July 29, 2021, 10:40-11:10 EDT,
Speaker: Thotsaporn "Aek" Thanatipanonda, Mahidol University
Title: The Card Guessing Game: A Generating Function Approach
Abstract: Consider a card guessing game with complete feedback, in which a deck of n cards, labeled 1, ..., n is riffle-shuffled once. With the goal to maximize the number of correct guesses, a player guesses the cards from the top of the deck, one at a time, under the optimal strategy until no cards remain. We provide the statistics (e.g. mean, variance, etc.) of the number of correct guesses.
Date: Thursday, July 29, 2021, 11:20-11:50 EDT,
Speaker: Sahoshi Chen, Chinese Academy of Sciences
Title: A Residue-Based Approach to Vanishing Sums
Abstract: Residue theorems have been used extensively to evaluate definite integrals and sums in combinatorics and special functions. In this talk, we will first give an overview of several methods for proving combinatorial identities, including the celebrated Wilf-Zeilberger method, and then present a new approach to proving and discovering vanishing sums based on a residue theorem due to Nicole in 1717. This is a joint work with Rong-Hua Wang.
Date: Thursday, July 29, 2021, 12:00-12:30 EDT,
Speaker: Aviezri Fraenkel, Weizmann Institute
Title: Patrolling the Border of a Striking Conjecture
Abstract: We discuss the following conjecture concerning disjointness and covering of integers: There is a unique system of sequences involving powers of 2 with distinct moduli, covering all integers. One of our main results is that the conjecture almost holds if powers of 2 are replaced by any power of a number greater than 2. In that case, two of the sequences necessarily intersect, but the others can be disjoint. When two sequences intersect, some numbers must be missing from the union of all sequences, since the reciprocals of all the moduli sum to 1 by density considerations. We study the structure of the complementary sequence.
Slides : this directory