Counting arithmetical structures on paths and cycles
Counting arithmetical structures on paths and cycles
复制标题
计算路径和循环上的算术结构
DOI：
10.1016/j.disc.2018.07.002
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
C. E. Valencia
中科院分区：
文献类型：
--
作者：
Benjamin Braun;Hugo Corrales;Scott Corry;L. García;D. Glass;N. Kaplan;Jeremy L. Martin;Gregg Musiker;C. E. Valencia
设G是一个有限连通图。G上的算术结构是一对正整数向量d， r，使得（diag (d)−a） r= 0，其中a是G的邻接矩阵。我们研究了路径图和循环图上算术结构的组合，以及相关的临界群（矩阵（diag (d)−a）的核的扭转部分）。对于路径，我们证明了算术结构是由加泰罗尼亚数枚举的，并得到了与投票序列相关的精细枚举结果。对于循环，我们证明了算术结构是用二项式系数2n−1 n−1来枚举的，并得到了与多集相关的精细枚举结果。此外，我们还确定了路径和环上所有算术结构的临界群。
Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d)− A) r= 0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d)− A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients 2 n− 1 n− 1, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.
影响因子：
0.6
作者：
Browning, T. D.;Elsholtz, C.
通讯作者：
Elsholtz, C.