Probabilistic parking functions (Q6133160)

Summary: 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 [Coll. Math. J. 52, No. 4, 254--264 (2021; Zbl 1505.97055)] by \textit{J. Carlson} et al. 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. Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.