(q,t)-Catalan numbers: gamma expansions, pattern avoidances, and the (−1)-phenomenon

Abstract

The aim of this paper is two-fold. We first prove several new interpretations of a kind of (q,t)-Catalan numbers along with their corresponding γ-expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain (−1)-phenomenon for each subset of permutations avoiding a single pattern of length three, and discuss their q-analogues utilizing the newly obtained q-γ-expansions, as well as the continued fraction of a quint-variate generating function due to Shin and the fourth author. Moreover, we enumerate the alternating permutations avoiding simultaneously two patterns, namely (2413,3142) and (1342,2431), of length four, and consider such (−1)-phenomenon for these two subsets as well.