论文标题
车轮和风扇图上的沙珀模型的组合方面
Combinatorial aspects of sandpile models on wheel and fan graphs
论文作者
论文摘要
我们研究了方向盘和风扇图上的沙珀模型的组合方面，从而寻求模型对这些家族的复发构型的两类射合特征。对于车轮图，我们在这些复发构型和周期图的子图集之间展示了两次射击，该图将配置的水平映射到子图的边缘数量。这种培训依赖于两种关键成分。第一个是考虑标准Abelian Sandpile模型（ASM）而不是ASM本身的随机变体的组成。第二个成分是从给定的复发状态到规范最小复发状态的映射，利用了与以前对ASM的相似思想在完整的两部分图和Ferrers图上。我们还表明，在带有$ 2N $顶点的车轮图上，具有$ n $级别的经常性状态的数量由中央Delannoy数字的第一个差异给出。最后，使用类似的工具，我们在扇形图上的ASM的一组复发配置与包含路径最正确的顶点的路径图的子图集之间展示了两者。我们证明，这些集合也与某些晶格路径相等，我们将其命名为Kimberling Paths，这是在整数序列的在线百科全书中的相应条目作者之后。
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.
