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Parking Functions, Multi-Shuffle, and Asymptotic Phenomena

Author Mei Yin

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Mei Yin
  • Department of Mathematics, University of Denver, Denver, CO, USA

Funding

Research partially supported by the University of Denver’s Faculty Research Fund 84688-145601.

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Mei Yin. Parking Functions, Multi-Shuffle, and Asymptotic Phenomena. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 18:1-18:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.AofA.2022.18

Abstract

Given a positive integer-valued vector u = (u_1, … , u_m) with u_1 < ⋯ < u_m, a u-parking function of length m is a sequence π = (π_1, … , π_m) of positive integers whose non-decreasing rearrangement (λ_1, … , λ_m) satisfies λ_i ≤ u_i for all 1 ≤ i ≤ m. We introduce a combinatorial construction termed a parking function multi-shuffle to generic u-parking functions and obtain an explicit characterization of multiple parking coordinates. As an application, we derive various asymptotic probabilistic properties of a uniform u-parking function of length m when u_i = cm+ib. The asymptotic scenario in the generic situation c > 0 is in sharp contrast with that of the special situation c = 0.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probability and statistics
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Parking function
  • Multi-shuffle
  • Asymptotic expansion
  • Abel’s multinomial theorem

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References

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