Abstract
Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case \(m=n\); we study here combinatorial and probabilistic aspects of this generalized case. We construct a family of bijections between parking functions \(\text {PF}(m, n)\) with m cars and n spots and spanning forests \(\mathscr {F}(n+1, n+1-m)\) with \(n+1\) vertices and \(n+1-m\) distinct trees having specified roots. This leads to a bijective correspondence between \(\text {PF}(m, n)\) and monomial terms in the associated Tutte polynomial of a disjoint union of \(n-m+1\) complete graphs. We present an identity between the “inversion enumerator” of spanning forests with fixed roots and the “displacement enumerator” of parking functions. The displacement is then related to the number of graphs on \(n+1\) labeled vertices with a fixed number of edges, where the graph has \(n+1-m\) disjoint rooted components with specified roots. We investigate various probabilistic properties of a uniform parking function, giving a formula for the law of a single coordinate. As a side result we obtain a recurrence relation for the displacement enumerator. Adapting known results on random linear probes, we further deduce the covariance between two coordinates when \(m=n\).
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The datasets generated during and/or analysed during the current study are available from the corresponding author Mei Yin on reasonable request.
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Acknowledgements
Mei Yin acknowledges helpful conversations with Lingjiong Zhu and constructive comments from Mitsuru Wilson.
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Richard Kenyon was supported by NSF DMS-1940932 and the Simons Foundation grant 327929. Mei Yin was supported by the University of Denver’s Faculty Research Fund 84688-145601 and Professional Research Opportunities for Faculty Fund 80369-145601.
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Richard Kenyon and Mei Yin contributed equally to the manuscript.
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Kenyon, R., Yin, M. Parking Functions: From Combinatorics to Probability. Methodol Comput Appl Probab 25, 32 (2023). https://doi.org/10.1007/s11009-023-10022-5
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DOI: https://doi.org/10.1007/s11009-023-10022-5