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Skeletal generalizations of Dyck paths, parking functions, and chip-firing games
- Publication Year :
- 2024
-
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.<br />Comment: 29 pages, 9 figures
- Subjects :
- Mathematics - Combinatorics
05A15, 05A19, 05C57
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2408.06923
- Document Type :
- Working Paper