The sandpile model on a bipartite graph, parallelogram polyominoes, and a q,t-Narayana polynomial

We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph Km,n in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a m × n rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. We define a collection of polynomials that we call q, t-Narayana polynomials, the generating functions of the bistatistic (area, parabounce) on the set of parallelogram polyominoes,
akin to Haglund's (area, hagbounce) bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. 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 discuss the proofs for numerous special cases. We also show a relationship between the q, t-Catalan polynomials and our bistatistic (area, parabounce) on a subset of parallelogram polyominoes.