Combinatorial aspects of sandpile models on wheel and fan graphs

TL;DR

This paper explores combinatorial properties of the sandpile model on wheel and fan graphs, establishing bijections with subgraphs and lattice paths, and linking recurrent configurations to well-known combinatorial sequences.

Contribution

It introduces new bijective characterisations of recurrent configurations on wheel and fan graphs using stochastic variants and minimal states, connecting these to classical combinatorial objects.

Findings

Bijection between recurrent configurations and subgraphs of cycle graphs.

Number of recurrent states with a given level relates to Delannoy numbers.

Recurrent configurations on fan graphs correspond to Kimberling paths.

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 configurations 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, using similar tools, we exhibit a bijection between the set of recurrent configurations of the ASM on fan graphs and the set of subgraphs of the path graph containing the right-most vertex of the path. We show that these sets are also equinumerous with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.