A weight-dependent inversion statistic and Catalan numbers

Combinatorics/Partitions Seminar

Meeting Details

For more information about this meeting, contact Kristin Berrigan, George Andrews, Ae Ja Yee, Matthew Katz.

Speaker: Michael Schlosser, University of Vienna

Abstract: We introduce a weight-dependent extension of the inversion statistic, a classical Mahonian statistic on permutations.This immediately gives us a new weight-dependent extension of n!. When we restrict to 312-avoiding permutations, our extension gives rise to a weight-dependent family of Catalan numbers, which happen to coincide with the weighted Catalan numbers that were previously introduced by Postnikov and Sagan by weighted enumeration of Dyck paths. While Postnikov and Sagan’s main focus was on the modulo 2 divisibility of the weighted Catalan numbers, we worked out further properties of these numbers that extend those of the classical case, such as their recurrence relation, their continued fraction, and Hankel determinants. We also discovered an intriguing closed form evaluation of the weighted Catalan numbers for a specific choice of weights. We further present bi-weighted Catalan numbers that generalize Garsia and Haiman’s q,t-Catalan numbers, and again satisfy remarkable properties. These are obtained by refining the weighted Catalan numbers by introducing an additional statistic, namely a weight-dependent extension of Haglund’s bounce statistic. This is joint work with Shishuo Fu (Chongqing University).

Room Reservation Information

Room Number: 106 McAllister

Date: 11/19/2019

Time: 11:15am - 12:05pm