Parallelogram polyominoes, the sandpile model on a bipartite graph, and a q, t-Narayana polynomial

In this talk I will highlight some results from a recent paper (arXiv:1208.0024) that was motived by a correspondence between bivincular permutation patterns and composition matrices. We study recurrent configurations of the sandpile model on the complete bipartite graph Km,n and show how they can be classified in terms of a class of polyominoes. A canonical toppling process on these recurrent states gives rise to a bounce path within the corresponding polyomino. This bounce path, in turn, gives rise to a polynomial that we call the q, tNarayana polynomial via a bistatistic on the polyominoes. This new polynomial brings with it a surprising link to the famous diagonal harmonics. We discuss this q, t-Narayana polynomial and its relation to the well-known q, t-Catalan polynomial.