Combinatorial aspects of sandpile models on wheel and fan graphs

Abstract

We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model's recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configuations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a given recurrent state to a canonical minimal recurrent state, exploiting similar ideas to previous studies of the ASM on complete bipartite graphs and Ferrers graphs. We also show that on the wheel graph with 2n vertices, the number of recurrent states with level n is given by the first differences of the central Delannoy numbers. Finally, we prove using similar tools that the recurrent configurations of the ASM on fan graphs are in bijection with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.