Set-valued tableaux and enumeration of special chip configurations on graphs

Chip-firing games on finite graphs provide a combinatorial analog of the theory of special divisors on algebraic curves. Each chip configuration determines two numbers: the degree and rank, analogous to the degree and dimension of a linear series on an algebraic curve. I will consider the enumerative problem: on a given finite graph, how many classes of chip configuration are there of a given degree and rank? I will state a complete solution to this problem in the case of a famous example—the “chains of loops” studied by Cools, Draisma, Payne, and Robeva. The solution has an intriguing form: it is a symmetric polynomial, with coefficients given by counting set-valued Young tableaux, evaluated on the lengths of the loops. I will discuss some intriguing parallels to algebraic geometry, in which the same set-valued tableaux arise in the cohomology class and Euler characteristic of Brill-Noether varieties, and speculate on potential generalizations to other families of graphs and to motives of Brill-Noether varieties.

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