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arXiv · 2006.09321

Interval parking functions

Emma Colaric·Ryan DeMuse·Jeremy L. Martin·Mei Yin
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Abstract

Interval parking functions (IPFs) are a generalization of ordinary parking functions in which each car is willing to park only in a fixed interval of spaces. Each interval parking function can be expressed as a pair $(a,b)$, where $a$ is a parking function and $b$ is a dual parking function. We say that a pair of permutations $(x,y)$ is \emph{reachable} if there is an IPF $(a,b)$ such that $x,y$ are the outcomes of $a,b$, respectively, as parking functions. Reachability is reflexive and antisymmetric, but not in general transitive. We prove that its transitive closure, the \emph{pseudoreachability order}, is precisely the bubble-sort order on the symmetric group $\Sym_n$, which can be expressed in terms of the normal form of a permutation in the sense of du~Cloux; in particular, it is isomorphic to the product of chains of lengths $2,\dots,n$. It is thus seen to be a special case of Armstrong's sorting order, which lies between the Bruhat and (left) weak orders.

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Emma Colaric, Ryan DeMuse, Jeremy L. Martin, Mei Yin. 2020-06-16. Interval parking functions. https://arxiv.org/abs/2006.09321

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