An area-depth symmetric $q,t$-Catalan polynomial

Abstract

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.

17 pages; v2 references added; v3 typos fixed, version to appear in Electronic J. Combinatorics

References in corpus (1)

• $e$-Positivity Results and Conjectures

Cited by in corpus (1)

• An area-bounce exchanging bijection on a large subset of Dyck paths