Skeletal generalizations of Dyck paths, parking functions, and chip-firing games
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929457876434944 |
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| author | Backman, Spencer Charbonneau, Cole Loehr, Nicholas A. Mullins, Patrick O'Connor, Mazie Warrington, Gregory S. |
| author_facet | Backman, Spencer Charbonneau, Cole Loehr, Nicholas A. Mullins, Patrick O'Connor, Mazie Warrington, Gregory S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06923 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Skeletal generalizations of Dyck paths, parking functions, and chip-firing games Backman, Spencer Charbonneau, Cole Loehr, Nicholas A. Mullins, Patrick O'Connor, Mazie Warrington, Gregory S. Combinatorics 05A15, 05A19, 05C57 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. |
| title | Skeletal generalizations of Dyck paths, parking functions, and chip-firing games |
| topic | Combinatorics 05A15, 05A19, 05C57 |
| url | https://arxiv.org/abs/2408.06923 |