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

We classify recurrent configurations of the sandpile model on the complete bipartite graph Km,n in which one designated vertex is a sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is an m×n rectangle. Several special types of recurrent configurations and their properties via this bijection are examined. For example, recurrent configurations whose sum of heights is minimal are shown to correspond to polyominoes of least area. Two other classes of recurrent configurations are shown to be related to bicomposition matrices, a matrix analogue of set partitions, and (2+2)-free partially ordered sets.A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. This path bounces off the external edges of the polyomino, and is reminiscent of Haglundʼs well-known bounce statistic for Dyck paths. We define a collection of polynomials that we call q,t-Narayana polynomials, defined to be the generating function of the bistatistic (area,parabounce) on the set of parallelogram polyominoes, akin to the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In doing so, we have extended a bistatistic of Egge et al. (2003) to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects.We conjecture the q,t-Narayana polynomials to be symmetric and prove this conjecture for numerous special cases. We also show a relationship between Haglundʼs (area,hagbounce) statistic on Dyck paths, and our bistatistic (area,parabounce) on a sub-collection of those parallelogram polyominoes living in a (n+1)×n rectangle.