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

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 $α$ 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.