Workshop on Enumerative Combinatorics 2021

Schedule

* Please note the timezone in Ireland is CET-1. *

2:00-2:05	Welcome
2:05-2:35	Ilse Fischer (Vienna, Austria)	Bijective proofs of alternating sign matrix theorems	SLIDES
2:40-2:50	Jehanne Dousse (Lyon, France)	Partition identities and representation theory	SLIDES
2:55-3:05	Angela Carnevale (NUIG, Ireland)	Odd diagrams of permutations	SLIDES
3:10-3:25	Coffee break
3:25-3:55	Peter McNamara (Bucknell, USA)	From Dyck paths to standard Young tableaux	SLIDES
4:00-4:10	Aoife Hennessy (WIT, Ireland)	Riordan arrays and Lattice paths	SLIDES
4:15-4:25	Elia Bisi (TU Vienna, Austria)	Sorting networks and staircase Young tableaux	SLIDES
4:25-4:30	Closing

Abstracts

Ilse Fischer: Bijective proofs of alternating sign matrix theorems
Abstract: Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, and their numbers are given by a simple product formula. For about 40 years now, combinatorialists have been trying to construct bijective proofs of these relations. We present the first bijective proof of the enumeration formula for alternating sign matrices and of the fact that alternating sign matrices are equinumerous with descending plane partitions. Our constructions rely on signed sets, sijections and related notions such as a generalization of the Garsia-Milne involution principle. The starting point for these constructions are known “computational” proofs, but the combinatorial point of view led to several drastic modifications. We also provide computer code where all of our constructions have been implemented. This is joint work with Matjaz Konvalinka.

Jehanne Dousse: Partition identities and representation theory
Abstract: A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. A partition identity is a theorem stating that for all n, the number of partitions of n satisfying some conditions equals the number of partitions of n satisfying some other conditions. We will give a quick overview of the fruitful interaction between partition identities and representation theory, from vertex operators to crystal bases.

Angela Carnevale: Odd diagrams of permutations
Abstract: The odd diagram of a permutation is a subset of the classical diagram defined by certain parity conditions. In this talk, I will focus on classes of permutations with the same odd diagram. In particular, we'll see that these classes enjoy remarkable properties related to pattern avoidance and to Bruhat order in symmetric groups. This is based on joint work with Francesco Brenti and Bridget Tenner.

Peter McNamara: From Dyck paths to standard Young tableaux
Abstract: Dyck paths and standard Young tableaux (SYT) are two of the most central sets in combinatorics. There is a well-known bijection between Dyck paths with 2n steps and SYT of shape (n,n). In recent work, we found nine other bijections between classes of Dyck paths and classes of SYT. I will present some of my favorites. This is joint work with Juan Gil, Jordan Tirrell and Michael Weiner.

Aoife Hennessy: Riordan arrays and Lattice paths
Abstract: This talk gives a brief introduction to the concept of the Riordan group, in particular the A and Z sequence relating to any Riordan array. We then use the A and Z sequences to illustrate a combinatorial interpretation of Riordan arrays in terms of weighted lattice paths.

Elia Bisi: Sorting networks and staircase Young tableaux
Abstract: The Edelman-Greene bijection is a correspondence between sorting networks and standard Young tableaux of staircase shape. After reviewing this mapping, I will present a new conjectural identity between generating series of these two sets of combinatorial objects. Joint work with Fabio D. Cunden, Shane Gibbons, and Dan Romik.

Organisers

Mark Dukes, Neil O'Connell, Robert Osburn.