Geodesic
Probabilistic parking functions
The electronic journal of combinatorics, Tome 30 (2023) no. 3
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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 Rodríguez. 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.
DOI : 10.37236/11649
Classification : 90B06, 05A19, 60C05

Irfan Durmić    ; Alex Han    ; Pamela E. Harris  1   ; Rodrigo Ribeiro    ; Mei Yin 

1 Williams College 
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     title = {Probabilistic parking functions},
     journal = {The electronic journal of combinatorics},
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Irfan Durmić; Alex Han; Pamela E. Harris; Rodrigo Ribeiro; Mei Yin. Probabilistic parking functions. The electronic journal of combinatorics, Tome 30 (2023) no. 3. doi: 10.37236/11649

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