Document Open Access Logo

Statistics of Parking Functions and Labeled Forests

Authors Stephan Wagner , Mei Yin

PDF
Thumbnail PDF

File

LIPIcs.AofA.2024.29.pdf
  • Filesize: 0.66 MB
  • 14 pages

Document Identifiers

Author Details

Stephan Wagner
  • Institute of Discrete Mathematics, TU Graz, Austria
  • Department of Mathematics, Uppsala University, Sweden
Mei Yin
  • Department of Mathematics, University of Denver, CO, USA

Funding

  • Wagner, Stephan: Supported by the Swedish research council (VR), grant 2022-04030.
  • Yin, Mei: Supported by the University of Denver’s Professional Research Opportunities for Faculty Fund 80369-145601.

Acknowledgements

The authors benefited from participation in the Workshop on Analytic and Probabilistic Combinatorics at the Banff International Research Station in November 2022 and also from participation in the 34th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms in Taipei in June 2023.

Cite As Get BibTex

Stephan Wagner and Mei Yin. Statistics of Parking Functions and Labeled Forests. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.AofA.2024.29

Abstract

In this paper we obtain some new results on the enumeration of parking functions and labeled forests. We introduce new statistics both for parking functions and for labeled forests that are connected to each other by means of a bijection. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Generating functions
Keywords
  • parking function
  • labeled forest
  • generating function
  • Pollak’s circle argument
  • bijection

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Philippe Chassaing and Jean-François Marckert. Parking functions, empirical processes, and the width of rooted labeled trees. Electron. J. Combin., 8(1):Research Paper 14, 19, 2001. URL: https://doi.org/10.37236/1558.
  2. Persi Diaconis and Angela Hicks. Probabilizing parking functions. Adv. in Appl. Math., 89:125-155, 2017. URL: https://doi.org/10.1016/j.aam.2017.05.004.
  3. Ömer Eğecioğlu and Jeffrey B. Remmel. Bijections for Cayley trees, spanning trees, and their q-analogues. J. Combin. Theory Ser. A, 42(1):15-30, 1986. URL: https://doi.org/10.1016/0097-3165(86)90004-X.
  4. Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. Google Scholar
  5. Dominique Foata and John Riordan. Mappings of acyclic and parking functions. Aequationes Math., 10:10-22, 1974. URL: https://doi.org/10.1007/BF01834776.
  6. António Guedes de Oliveira and Michel Las Vergnas. Parking functions and labeled trees. Sém. Lothar. Combin., 65:Art. B65e, 10, 2010/12. Google Scholar
  7. Frank Harary and Edgar M. Palmer. Graphical enumeration. Academic Press, New York-London, 1973. Google Scholar
  8. Richard Kenyon and Mei Yin. Parking functions: from combinatorics to probability. Methodol. Comput. Appl. Probab., 25(1):Paper No. 32, 30, 2023. URL: https://doi.org/10.1007/s11009-023-10022-5.
  9. Alan G. Konheim and Benjamin Weiss. An occupancy discipline and applications. SIAM J. Appl. Math., 14:1266-1274, 1966. Google Scholar
  10. Marie-Louise Lackner and Alois Panholzer. Runs in labelled trees and mappings. Discrete Math., 343(9):111990, 16, 2020. URL: https://doi.org/10.1016/j.disc.2020.111990.
  11. C. L. Mallows and John Riordan. The inversion enumerator for labeled trees. Bull. Amer. Math. Soc., 74:92-94, 1968. URL: https://doi.org/10.1090/S0002-9904-1968-11888-9.
  12. John Riordan. Ballots and trees. J. Combinatorial Theory, 6:408-411, 1969. Google Scholar
  13. Heesung Shin and Jiang Zeng. A bijective enumeration of labeled trees with given indegree sequence. J. Combin. Theory Ser. A, 118(1):115-128, 2011. URL: https://doi.org/10.1016/j.jcta.2010.07.001.
  14. Richard P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. URL: https://doi.org/10.1017/CBO9780511609589.
  15. Richard P. Stanley and Jim Pitman. A polytope related to empirical distributions, plane trees, parking functions, and the associahedron. Discrete Comput. Geom., 27(4):603-634, 2002. URL: https://doi.org/10.1007/s00454-002-2776-6.
  16. Richard P. Stanley and Mei Yin. Some enumerative properties of parking functions, 2023. arXiv: 2306.08681. Google Scholar
  17. Catherine H. Yan. Parking functions. In Handbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), pages 835-893. CRC Press, Boca Raton, FL, 2015. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact