Document Open Access Logo

Formulas and Asymptotics of Hypergraph Catalan Numbers

Author Eva-Maria Hainzl

PDF

File

Thumbnail PDF
PDF
LIPIcs.AofA.2026.20.pdf
  • Filesize: 0.72 MB
  • 15 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
Keywords
  • generating functions
  • trees
  • tree walks
  • catalan numbers

Metrics

Abstract

Tree walks are a class of closed walks on a complete graph constrained to span trees. They appear in the computation of moments of the spectral measure of various random matrix models, most prominently in the spectral distribution of random graphs. In this work, we focus on a special subclass called k-tours, which were introduced by Gunnells [Gunnels, 2021] after studying another random matrix model. They are enumerated by the so-called hypergraph Catalan numbers c_n^(k). Gunnells conjectured an asymptotic formula for c_n^(k), which we confirm through an alternative approach to their enumeration. As it turns out, the asymptotic growth is governed by the number of k-tours on star-like trees.

Cite As Get BibTex

Eva-Maria Hainzl. Formulas and Asymptotics of Hypergraph Catalan Numbers. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.20

Author Details

Eva-Maria Hainzl
  • LIX, École polytechnique, CNRS, Institut Polytechnique de Paris, France

Funding

  • Hainzl, Eva-Maria: This work was partially founded by the Austrian Science Foundation FWF, projects P 35016, F 100203 and ANR LOUCCOUM.

Acknowledgements

The author was informed by Paul Gunnells that Mario DeFranco independently found the formula for the hypergraph Catalan numbers and computed their asymptotic expansion during his employment at UMass, but he left academia before formally writing up his results. She would also like to thank her supervisor Michael Drmota for the helpful discussions.

References

  1. Michel Bauer and Olivier Golinelli. Random incidence matrices: Moments of the spectral density. Journal of Statistical Physics, 103:301-337, 2001. URL: https://doi.org/10.1023/A:1004879905284.
  2. Charles Bordenave, Marc Lelarge, and Justin Salez. The rank of diluted random graphs. The Annals of Probability 39(3), pages 1097-1121, 2011. Google Scholar
  3. Charles Bordenave, A. Sen, and B. Virág. Mean quantum percolation. Journal of the European Mathematical Society 19, pages 3679-3707, 2017. Google Scholar
  4. Nathanaël Enriquez and Laurent Ménard. Spectra of large diluted but bushy random graphs. Random Structures & Algorithms, 49, October 2013. URL: https://doi.org/10.1002/rsa.20618.
  5. Ira M. Gessel and Richard P. Stanley. Algebraic enumeration, pages 1021-1061. MIT Press, Cambridge, MA, USA, 1996. Google Scholar
  6. Paul Gunnells. Hypergraph matrix models and generating functions. Combinatorics and Number Theory, 13:149-178, July 2024. URL: https://doi.org/10.2140/cnt.2024.13.149.
  7. Paul Gunnels. Generalized catalan numbers from hypergraphs. Electronic Journal of Combinatorics, 28, 2021. Google Scholar
  8. Eva-Maria Hainzl and Élie de Panafieu. Tree Walks and the Spectrum of Random Graphs. In Cécile Mailler and Sebastian Wild, editors, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024), volume 302 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:15, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.AofA.2024.11.
  9. A. Yavuz Oruç. On number of partitions of an integer into a fixed number of positive integers. Journal of Number Theory, 159:355-369, 2016. URL: https://doi.org/10.1016/j.jnt.2015.06.023.
  10. Justin Salez. Every totally real algebraic integer is a tree eigenvalue. Journal of Combinatorial Theory, Series B, pages 249-256, 2015. URL: https://doi.org/10.1016/J.JCTB.2014.09.001.
  11. Justin Salez. Spectral atoms of unimodular random trees. Journal of the European Mathematical Society 22, pages 345-363, 2020. Google Scholar
  12. R. Stanley. Catalan numbers. Cambridge University Press, 2015. Google Scholar
  13. Inna Zakharevich. A generalization of wigner’s law. Communications in Mathematical Physics, 268:403-414, 2006. URL: https://doi.org/10.1007/s00220-006-0074-5.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact