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arXiv · 2410.22232

Primeness of generalized parking functions

Sam Armon·Joanne Beckford·Dillon Hanson·Naomi Krawzik·Olya Mandelshtam·Lucy Martinez·Catherine Yan
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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.

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Sam Armon, Joanne Beckford, Dillon Hanson, Naomi Krawzik, Olya Mandelshtam, Lucy Martinez, Catherine Yan. 2024-10-29. Primeness of generalized parking functions. https://arxiv.org/abs/2410.22232

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