Counting Parking Sequences and Parking Assortments Through Permutations

Abstract: Parking sequences (a generalization of parking functions) are defined by specifying car lengths and requiring that a car attempts to park in the first available spot after its preference. If it does not fit there, then a collision occurs and the car fails to park. In contrast, parking assortments generalize parking sequences (and parking functions) by allowing cars (also of assorted lengths) to seek forward from their preference to identify a set of contiguous unoccupied spots in which they fit. We consider both parking sequences and parking assortments and establish that the number of preferences resulting in a fixed parking order $\sigma$ is related to the lengths of cars indexed by certain subsequences in $\sigma$. The sum of these numbers over all parking orders (i.e. permutations of $[n]$) yields new formulas for the total number of parking sequences and of parking assortments.