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arXiv · 2601.20770

On statistics of prime parking functions, {\L}ukasiewicz paths, and quasisymmetric functions

Pamela E. Harris·Selvi Kara·Erin McNicholas·Kathryn Nyman·Mei Yin
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Abstract

We recall that a parking function of length $n+1$ is said to be prime if removing any instance of 1 yields a parking function of length $n$. In this article, we study prime parking functions from multiple lenses. We derive an explicit formula for the average value of the total displacement of prime parking functions. We present a formula for the displacement-enumerator of prime parking functions that involves a sum over {\L}ukasiewicz paths. We describe the one-to-one correspondence between parking functions and labeled{\L}ukasiewicz paths via Dyck paths. We introduce the concept of $\ell$-forward differences and use this as a vehicle for examining ties, ascents, and descents in prime parking functions. We establish a link between Schur functions corresponding to the partition $(i,1^{n-i})$ and fundamental quasisymmetric functions indexed by prime parking function tie sets of size $n-i.$

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Pamela E. Harris, Selvi Kara, Erin McNicholas, Kathryn Nyman, Mei Yin. 2026-01-28. On statistics of prime parking functions, {\L}ukasiewicz paths, and quasisymmetric functions. https://arxiv.org/abs/2601.20770

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