Probabilistic Parking Functions – 2022

with Irfan Durmic (Williams College), Alex Han (Williams College), Pamela E. Harris (University of Wisconsin-Milwaukee) and Mei Yin (University of Denver)


We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper “Parking Functions: Choose your own adventure,” (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodr\’iguez.
Among our results, we prove that the probability of obtaining a parking function, from a length $n$ preference vector, is independent of the probabilistic parameter $p$.
We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter $p$. Of special interest is when $p=1/2$, where we demonstrate a sharp transition in some parking statistics.
We also present several interesting combinatorial consequences of the parking protocol.
In particular, we provide a combinatorial interpretation for the array described in OEIS A220884 as the expected number of preference sequences with a particular property related to occupied parking spots, which solves an open problem of Novelli and Thibon posed in 2020 (arXiv:1209.5959). Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.

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