Skeletal generalizations of Dyck paths, parking functions, and chip-firing games
| Autor: | Backman, Spencer, Charbonneau, Cole, Loehr, Nicholas A., Mullins, Patrick, O'Connor, Mazie, Warrington, Gregory S. |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 29 pages, 9 figures |
| Databáze: | arXiv |
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