Title: Rational parking functions and $(m, n)$-invariant sets Show Chinese title

Title: 有理停车函数和$(m, n)$-不变集

Abstract: 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 $\Delta \subset \mathbb{Z}$ such that $\Delta + m \subset \Delta$ and $\Delta + n \subset \Delta$. 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 $\Delta$. 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. Show Chinese abstract

Abstract: 一个$(m, n)$-停车函数可以被描述为函数$f:[n] \to [m]$，使得通过重新排列$f$的值所得到的分区适合于一条直角边长分别为$m$和$n$的直角三角形内。 McCammond、Thomas 和 Williams 的近期工作定义了$[m]^n$中单词在$\mathbb{R}^n$上的作用。他们表明，有理停车函数恰好是那些在这个作用下具有不动点的单词。 一个$(m, n)$-不变集是一个集合$\Delta \subset \mathbb{Z}$，使得$\Delta + m \subset \Delta$和$\Delta + n \subset \Delta$。 在这项工作中，我们通过从$\Delta$中去掉第$j$个$m$-生成元，定义了$[m]^n $上的单词在$(m, n)$-不变集上的作用。 我们证明这种作用同样刻画了$(m, n)$-parking 函数。 进一步地，我们证明每个$(m, n)$-不变集由唯一的单调 parking 函数固定。 通过关联作用于 $\mathbb{R}^m$ 和作用于 $(m, n)$-不变集上的操作，我们证明了 $\mathbb{R}^m$ 中能被停车函数固定的点的集合是单调停车函数固定的点的集合的并集。 当 $\gcd(m, n) =1$ 成立时，我们刻画了定义在 $\mathbb{R}^m$ 上的作用的周期点集合，并且证明了 Gorsky、Mazin 和 Vazirani 提出的反转 Pak-Stanley 映射的算法在有限步内收敛。