2025/02/28 by Nelson, Garrett · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.00181
An (m, n)-parking function can be characterized as function f:[n] → [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 ℝn. 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⊂ ℤ such that Δ+ m ⊂ Δ and Δ+ n ⊂ Δ. In this work we define an action of words in [m]n on (m, n)-invariant sets by removing the jth 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 ℝm and on (m, n)-invariant sets we prove that the set of all the points in ℝm 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 ℝm and show that the algorithm reversing the Pak-Stanley map proposed by Gorsky, Mazin, and Vazirani converges in a finite amount of steps.