UW Combinatorics and Geometry Seminar: Spencer Daugherty

The $q,t$-Catalan numbers can be described elegantly in terms of pairs of statistics on Dyck paths: area and bounce, or area and dinv. Using bijective and recursive methods, we prove new expressions of the $q,t$-Catalan numbers in terms of pairs of statistics on other Catalan objects such as subsets of permutations, noncrossing partitions, unit interval orders, and certain Young tableaux. Special attention will be given to two different formulations using the disorder statistic on permutations. We also study the relationship of our statistics with the zeta map on Dyck paths and discuss implications for the $q,t$-Catalan symmetry problem.