Title: A weight-dependent inversion statistic and Catalan numbers
Speaker: Michael Schlosser (University of Vienna, Austria)
Date: 06 November 2020
Time: 3 pm
Venue: Microsoft Teams (online)
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 $n!$.
By restricting to $312$-avoiding permutations our extension happens
to coincide with the weighted Catalan numbers that were considered
by Flajolet in his combinatorial study of continued fractions.
We show that for a specific choice of weights the weighted
Catalan numbers factorize into a closed form, hereby yielding a new
$q$-analogue of the Catalan numbers, different from
those considered by MacMahon, by Carlitz, or by Andrews.
We further refine the weighted Catalan numbers by introducing
an additional statistic, namely a weight-dependent extension of
Haglund’s bounce statistic, and obtain a new family of bi-weighted
Catalan numbers that generalize Garsia and Haiman’s $q,t$-Catalan
numbers and appear to satisfy remarkable properties.
This is joint work with Shishuo Fu.