On statistics of prime parking functions, Łukasiewicz paths, and quasisymmetric functions

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 Łukasiewicz paths. We describe the one-to-one correspondence between parking functions and labeledŁukasiewicz 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 $n-i.$

💡 Research Summary

This paper investigates prime parking functions—those parking functions of length $n!+!1$ that remain a valid parking function of length $n$ after the removal of any occurrence of the entry 1—from several complementary perspectives. The authors first exploit the permutation invariance of the set $PPF_{n+1}$ to show that all coordinates have the same marginal distribution, reducing the problem of computing the average total displacement to the computation of $\mathbb{E}