Combinatorics Seminar

16:00-17:00

Speaker: Nikolaos Pantelidis

Title: On the Riordan Group and Riordan quasi-involutions

Abstract: 

A Riordan array is a lower triangular infinite matrix, constructed by a pair of formal power series in such a way that each of the columns is generated by them, and the entries of the matrix can be determined by a recursive formula. Objects that can be expressed by a sequence of integers are used to construct Riordan arrays, such as Fibonacci or Catalan numbers.

Riordan arrays have been researched since the early 1990s and it is a growing field. Applications of them have been found in many areas of computing such as algorithm analysis, error correcting codes, wireless communications, along with scientific areas beyond the borders of Mathematics as parts of their theory and techniques have been successfully applied in Molecular Biology for RNA secondary structure enumeration and Chemistry.

In this talk, we are going to present a brief introduction to the related theory, focusing on some of our current research on the algebraic structure of the main group of Riordan arrays. We discuss conditions for Riordan elements of order 2, generalizations of them and relations between already known Riordan subgroups. Additionally, we present a type of special Riordan elements, called quasi-involutions, a factorization theorem and links to other topics.

17:00-18:00

Speaker: Jianping Pan

Title: Crystal for stable Grothendieck polynomials

Abstract: 

The Grothendieck polynomials arise from enumerative geometry, as they can be used to calculate the intersection numbers for the Flag varieties. We introduce a crystal on decreasing factorizations of 321-avoiding elements of the 0-Hecke monoid, whose generating functions are the stable Grothendieck polynomials. This crystal is a K-theoretic generalization of the Morse-Schilling crystal on decreasing factorizations in the symmetric group. We prove that it intertwines with the crystal on set-valued tableaux introduced by Monical, Pechenik, and Scrimshaw (through the residue map). We also define a new insertion algorithm that intertwines with our crystal, with surprising connections to the Hecke insertion algorithm and the uncrowding algorithm for set-valued tableaux.
 
This talk is based on joint work arXiv:1911.08732 with Jennifer Morse, Wencin Poh, and Anne Schilling.