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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.00181 |
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| _version_ | 1866912253200039936 |
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| author | Nelson, Garrett |
| author_facet | Nelson, Garrett |
| contents | An $(m, n)$-parking function can be characterized as function $f:[n] \to [m]$ such that the partition obtained by reordering the values of $f$ fits inside a right triangle with legs of length $m$ and $n$. Recent work by McCammond, Thomas, and Williams define an action of words in $[m]^n$ on $\mathbb{R}^n$. They show that rational parking functions are exactly the words that admit fixed points under that action. An $(m, n)$-invariant set is a set $Δ\subset \mathbb{Z}$ such that $Δ+ m \subset Δ$ and $Δ+ n \subset Δ$. In this work we define an action of words in $[m]^n $ on $(m, n)$-invariant sets by removing the $j$th $m$-generator from $Δ$. We show this action also characterizes $(m, n)$-parking functions. Further we show that each $(m, n)$-invariant set is fixed by a unique monotone parking function. By relating the actions on $\mathbb{R}^m$ and on $(m, n)$-invariant sets we prove that the set of all the points in $\mathbb{R}^m$ that can be fixed by a parking function is a union of points fixed by monotone parking functions. In the case when $\gcd(m, n) =1$ we characterize the set of periodic points of the action defined on $\mathbb{R}^m$ and show that the algorithm reversing the Pak-Stanley map proposed by Gorsky, Mazin, and Vazirani converges in a finite amount of steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00181 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational parking functions and $(m, n)$-invariant sets Nelson, Garrett Combinatorics An $(m, n)$-parking function can be characterized as function $f:[n] \to [m]$ such that the partition obtained by reordering the values of $f$ fits inside a right triangle with legs of length $m$ and $n$. Recent work by McCammond, Thomas, and Williams define an action of words in $[m]^n$ on $\mathbb{R}^n$. They show that rational parking functions are exactly the words that admit fixed points under that action. An $(m, n)$-invariant set is a set $Δ\subset \mathbb{Z}$ such that $Δ+ m \subset Δ$ and $Δ+ n \subset Δ$. In this work we define an action of words in $[m]^n $ on $(m, n)$-invariant sets by removing the $j$th $m$-generator from $Δ$. We show this action also characterizes $(m, n)$-parking functions. Further we show that each $(m, n)$-invariant set is fixed by a unique monotone parking function. By relating the actions on $\mathbb{R}^m$ and on $(m, n)$-invariant sets we prove that the set of all the points in $\mathbb{R}^m$ that can be fixed by a parking function is a union of points fixed by monotone parking functions. In the case when $\gcd(m, n) =1$ we characterize the set of periodic points of the action defined on $\mathbb{R}^m$ and show that the algorithm reversing the Pak-Stanley map proposed by Gorsky, Mazin, and Vazirani converges in a finite amount of steps. |
| title | Rational parking functions and $(m, n)$-invariant sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.00181 |