Enumerating Vector Parking Functions and their Outcomes Based on Specified Lucky Cars

TL;DR

This paper extends the enumeration of parking function outcomes to vector parking functions with fixed lucky cars, providing formulas and counts based on lucky cars and spot capacities, advancing combinatorial understanding.

Contribution

It introduces enumeration methods for vector parking functions with fixed lucky cars and counts outcomes based on lucky cars and spot capacities.

Findings

Enumerated outcomes of vector parking functions with fixed lucky cars.

Derived formulas for counting parking functions by lucky cars and spots.

Extended classical parking function enumeration to vector variants.

Abstract

In a parking function, a car is considered lucky if it is able to park in its preferred spot. Extending work of Harris and Martinez, we enumerate outcomes of parking functions with a fixed set of lucky cars. We then consider a generalization of parking functions known as vector parking functions or $\boldsymbol{u}$-parking functions, in which a nonnegative integer capacity is given to each parking spot in the street. With certain restrictions on $\boldsymbol{u}$, we enumerate outcomes of $\boldsymbol{u}$-parking functions with a fixed set of lucky cars or with a fixed number of lucky cars. We also count outcomes according to which spots contain lucky cars, and give formulas for enumerating $\boldsymbol{u}$-parking functions themselves according to their set of lucky cars.