Interval parking functions

TL;DR

Interval parking functions extend traditional parking functions by restricting each car to a fixed interval, revealing a new order structure related to bubble-sort and Armstrong's sorting order.

Contribution

This paper characterizes the reachability relation among interval parking functions and links it to the bubble-sort order and Armstrong's sorting order, providing new insights into their structure.

Findings

Reachability is reflexive and antisymmetric but not transitive.

The transitive closure, pseudoreachability, equals the bubble-sort order.

Pseudoreachability is isomorphic to a product of chains of lengths 2 to n.

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.