Primeness of generalized parking functions

TL;DR

This paper extends the concept of primeness from classical parking functions to three generalized forms, providing explicit enumeration formulas for prime cases in each generalization.

Contribution

It introduces the notion of primeness for vector, (p,q)-, and two-dimensional vector parking functions and derives explicit formulas for counting prime functions in these categories.

Findings

Explicit formulas for prime vector parking functions in arithmetic progressions

Enumeration of prime (p,q)-parking functions with specific parameters

Counting prime two-dimensional vector parking functions with affine weight matrices

Abstract

Classical parking functions are a generalization of permutations that appear in many combinatorial structures. Prime parking functions are indecomposable components such that any classical parking function can be uniquely described as a direct sum of prime ones. In this article, we extend the notion of primeness to three generalizations of classical parking functions: vector parking functions, $(p,q)$-parking functions, and two-dimensional vector parking functions. We study their enumeration by obtaining explicit formulas for the number of prime vector parking functions when the vector is an arithmetic progression, prime $(p,q)$-parking functions, and prime two-dimensional vector parking functions when the weight matrix is an affine transformation of the coordinates.