2020/06/16 by Emma Colaric, Ryan DeMuse, Colaric, Emma +5 · 4 citations
Computer Science · Engineering · Mathematics · #05A05 #05A15 #06A07 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2006.09321
openalex publication_date 2020/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 pseudoreachability order, is precisely the bubble-sort order on the symmetric group \Symn, 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,…,n. It is thus seen to be a special case of Armstrong's sorting order, which lies between the Bruhat and (left) weak orders.