A Colorful Way to Park: An Introduction to Exact $k$-Typed Parking Functions

TL;DR

This paper introduces exact $k$-typed parking functions, a new variant of classical parking functions, establishing their unique correspondence with parking configurations and exploring their relation to other combinatorial objects.

Contribution

It defines exact $k$-typed parking functions and proves their one-to-one correspondence with parking configurations, expanding the combinatorial understanding of parking functions.

Findings

Each exact $k$-typed parking function corresponds uniquely to a parking configuration.

The set of all exact $k$-TPFs leading to the same configuration forms a disjoint subset.

Parking permutations of an exact $k$-TPF relate to other combinatorial objects.

Abstract

Parking functions are tuples that describe the parking of $M$ cars on a street with $M$ parking spots. In this paper, we define exact $k$-typed parking functions ($k$-TPFs) to be a variant of classical parking functions. We then establish that every exact $k$-TPF $\alpha$ of length $M$, corresponds to a unique parking configuration $C$. We observe that the collection of all exact $k$-TPFs which result in the same configuration form a disjoint subset of all exact $k$-TPFs. Lastly, we conclude by showing how parking permutations of an exact $k$-TPF can be related to other combinatorial objects.