Skip to main content
SpringerLink
Log in

Universal Asymptotic Properties of Positive Functional Equations with One Catalytic Variable

  • Original Research Article
  • Published:
La Matematica Aims and scope Submit manuscript

Abstract

Functional equations with one catalytic variable appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions, the dominant singularity of the solution has a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square root singularity appears, and non-linear catalytic equations, where we—usually—have a singularity of type 3/2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In [6], an additional additive polynomial term \(F_0(z,u)\) is considered, however, by substituting M(z, u) by \(M(z,u)-F_0(z,u)\) the seemingly more general case can be reduced to the case (4).

References

  1. Tutte, W.T.: A census of planar maps. Can. J. Math. 15, 249–271 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banderier, C., Flajolet, P.: Basic analytic combinatorics of directed lattice paths. Theor. Comput. Sci. 281(1–2), 37–80 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Prodinger, H.: The kernel method: a collection of examples. Sémin. Lothar. Comb. 50, 50–19 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Asinowski, A., Bacher, A., Banderier, C., Gittenberger, B.: Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata. Algorithmica 82(3), 386–428 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brown, W.G., Tutte, W.T.: On the enumeration of rooted non-separable planar maps. Can. J. Math. 16, 572–577 (1964). https://doi.org/10.4153/CJM-1964-058-7

    Article  MathSciNet  MATH  Google Scholar 

  6. Bousquet-Mélou, M., Jehanne, A.: Polynomial equations with one catalytic variable, algebraic series and map enumeration. J. Comb. Theory, Ser. B 96(5), 623–672 (2006). https://doi.org/10.1016/j.jctb.2005.12.003

    Article  MathSciNet  MATH  Google Scholar 

  7. Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3(2), 216–240 (1990). https://doi.org/10.1137/0403019

    Article  MathSciNet  MATH  Google Scholar 

  8. Bostan, A., Chyzak, F., Notarantonio, H., El Din, M.S.: Algorithms for discrete differential equations of order 1. In: ISSAC ’22—Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, pp. 101–110. ACM, New York (2022)

  9. Bostan, A., Notarantonio, H., Din, M.S.E.: Fast Algorithms for Discrete Differential Equations (2023)

  10. Drmota, M., Noy, M., Yu, G.-R.: Universal singular exponents in catalytic variable equations. J. Comb. Theory, Ser. A 185, 33 (2022). https://doi.org/10.1016/j.jcta.2021.105522. Id/No 105522

  11. Banderier, C., Drmota, M.: Formulae and asymptotics for coefficients of algebraic functions. Comb. Probab. Comput. 24(1), 1–53 (2015). https://doi.org/10.1017/S0963548314000728

    Article  MathSciNet  MATH  Google Scholar 

  12. Buchacher, M., Kauers, M.: Inhomogeneous restricted lattice walks. Sém. Lothar. Combin. 82B, 75–12 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Notarantonio, H., Yurkevich, S.: Effective algebraicity for solutions of systems of functional equations with one catalytic variable. arXiv Preprint (2023). arXiv:2211.07298

  14. Drmota, M., Panagiotou, K.: A central limit theorem for the number of degree-\(k\) vertices in random maps. Algorithmica 66(4), 741–761 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Drmota, M., Yu, G.-R.: The Number of Double Triangles in Random Planar Maps. In: Fill, J.A., Ward, M.D. (eds.) 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 110, pp. 19–11918. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2018). https://doi.org/10.4230/LIPIcs.AofA.2018.19

  16. Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Modern Birkhäuser Classics, p. 721. Birkhäuser/Springer Basel AG, Basel (1986). https://doi.org/10.1007/978-3-0348-5097-1. Translated from the German original by John Stillwell, [2012] reprint of the 1986 edition

  17. Drmota, M.: Random trees. an interplay between combinatorics and probability. Wien: Springer (2009). https://doi.org/10.1007/978-3-211-75357-6

  18. Kaup, L., Kaup, B.: Holomorphic Functions of Several Variables, p. 349. Walter de Gruyter & Co., Berlin (1983)

    Book  MATH  Google Scholar 

  19. Bender, E.A.: An asymptotic expansion for the coefficients of some formal power series. J. London Math.So 9, 451–458 (1975)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referees for their careful reading and their valuable comments for improving the presentation.

Funding

This study was supported by FWF (Grant Nos. F50-02, F55-02, and P35016).

Author information

Author notes

  1. Michael Drmota and Eva-Maria Hainzl have contributed equally to this work.

Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstrasse 8-10, Vienna, 1040, Austria

    Michael Drmota & Eva-Maria Hainzl

Authors
  1. Michael Drmota
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eva-Maria Hainzl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eva-Maria Hainzl.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Data Availability

Upon request, the corresponding author can share their messy Sage worksheet and their Matlab computations and plots per mail.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Drmota, M., Hainzl, EM. Universal Asymptotic Properties of Positive Functional Equations with One Catalytic Variable. La Matematica 2, 692–742 (2023). https://doi.org/10.1007/s44007-023-00063-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s44007-023-00063-0

Keywords

Search

Navigation