Combinatorial aspects of sandpile models on wheel and fan graphs

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