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The sandpile model on the complete split graph, Motzkin words, and tiered parking functions
Journal of Combinatorial Theory Series A ( IF 1.1 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.jcta.2021.105418
Mark Dukes

We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of the recurrent states in terms of a new type of parking function that we call a tiered parking function. These parking functions are characterised by assigning a tier (or colour) to each of the cars, and specifying how many cars of a lower-tier one wishes to have parked before them. We also enumerate the different sets of recurrent configurations studied in this paper, and in doing so derive a formula for the number of spanning trees of the complete split graph that uses a bijective Prüfer code argument.



中文翻译:

完整拆分图上的沙堆模型,Motzkin单词和分层停车功能

我们在完整的分裂图中对Abelian沙堆模型(ASM)的递归状态进行分类。有两种不同的情况要考虑,这取决于完整拆分图中汇点顶点的位置。递减状态的这种表征是根据Motzkin单词,也可以根据组合项链来表征。我们还通过一种新型的停车功能(称为分层停车功能)来表征重复状态。这些停车功能的特点是为每辆车分配了一个等级(或颜色),并指定希望在其之前停放多少辆较低等级的车。我们还列举了本文研究的不同组递归配置,并以此推导了使用双射Prüfer代码参数的完整拆分图的生成树数量的公式。

更新日期:2021-02-01
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