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arXiv · 2408.06923

Skeletal generalizations of Dyck paths, parking functions, and chip-firing games

Spencer Backman·Cole Charbonneau·Nicholas A. Loehr·Patrick Mullins·Mazie O'Connor·Gregory S. Warrington
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Abstract

For $0\leq k\leq n-1$, we introduce a family of $k$-skeletal paths which are counted by the $n$-th Catalan number for each $k$, and specialize to Dyck paths when $k=n-1$. We similarly introduce $k$-skeletal parking functions which are equinumerous with the spanning trees on $n+1$ vertices for each $k$, and specialize to classical parking functions for $k=n-1$. The preceding constructions are generalized to paths lying in a trapezoid with base $c > 0$ and southeastern diagonal of slope $1/m$; $c$ and $m$ need not be integers. We give bijections among these families when $k$ varies with $m$ and $c$ fixed. Our constructions are motivated by chip firing and have connections to combinatorial representation theory and tropical geometry.

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Spencer Backman, Cole Charbonneau, Nicholas A. Loehr, Patrick Mullins, Mazie O'Connor, Gregory S. Warrington. 2024-08-13. Skeletal generalizations of Dyck paths, parking functions, and chip-firing games. https://arxiv.org/abs/2408.06923

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