Subset Parking Functions

Abstract

A parking function (c 1 , . . . , c n ) can be viewed as having n cars trying to park on a one-way street with n parking spots, where car i tries to park in spot c i , and otherwise he parks in the leftmost available spot after c i . Another way to view this is that each car has a set C i of "acceptable" parking spots, namely C i = [c i , n], and that each car tries to park in the leftmost available spot that they find acceptable.Motivated by this, we define a subset parking function (C 1 , . . . , C n ), with each C i a subset of {1, . . . , n}, by having the ith car try to park in the leftmost available element of C i . We further generalize this idea by restricting our sets to be of size k, intervals, and intervals of length k. In each of these cases we provide formulas for the number of such parking functions.