Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

Abstract

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.

References in corpus (2)

• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial

Cited by in corpus (5)

• The Schröder case of the generalized Delta conjecture
• The Abelian sandpile model on Ferrers graphs -- A classification of recurrent configurations
• The sandpile model on the complete split graph, Motzkin words, and tiered parking functions
• Combinatorics of Labelled Parallelogram polyominoes
• Perfect powers in Catalan and Narayana numbers