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arXiv · 1701.06377

Counting Arithmetical Structures on Paths and Cycles

Benjamin Braun·Hugo Corrales·Scott Corry·Luis David García Puente·Darren Glass·Nathan Kaplan·Jeremy L. Martin·Gregg Musiker·Carlos E. Valencia
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Abstract

Let $G$ be a finite, simple, connected graph. An arithmetical structure on $G$ is a pair of positive integer vectors $\mathbf{d},\mathbf{r}$ such that $(\mathrm{diag}(\mathbf{d})-A)\mathbf{r}=0$, 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 cokernels of the matrices $(\mathrm{diag}(\mathbf{d})-A)$). 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 $\binom{2n-1}{n-1}$, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

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Benjamin Braun, Hugo Corrales, Scott Corry, Luis David García Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, Carlos E. Valencia. 2017-01-23. Counting Arithmetical Structures on Paths and Cycles. https://doi.org/10.1016/j.disc.2018.07.002

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