Document Open Access Logo

Asymptotic Expansions for Sub-Critical Lagrangean Forms

Authors Hsien-Kuei Hwang , Mihyun Kang , Guan-Huei Duh

PDF
Thumbnail PDF

File

LIPIcs.AofA.2018.29.pdf
  • Filesize: 0.52 MB
  • 13 pages

Document Identifiers

Author Details

Hsien-Kuei Hwang
  • Institute of Statistical Science, Academia Sinica, Taiwan
Mihyun Kang
  • Institute of Discrete Mathematics, TU Graz, Austria
Guan-Huei Duh
  • Institute of Statistical Science, Academia Sinica, Taiwan

Funding

  • Hwang, Hsien-Kuei: Partially supported by the Ministry of Science and Technology (Taiwan) under the Grant MOST 104-2923-M-009-006
  • Kang, Mihyun: Partially supported by the Austrian Science Fund (FWF) under the Grant P2309-N35

Cite AsGet BibTex

Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh. Asymptotic Expansions for Sub-Critical Lagrangean Forms. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.AofA.2018.29

Abstract

Asymptotic expansions for the Taylor coefficients of the Lagrangean form phi(z)=zf(phi(z)) are examined with a focus on the calculations of the asymptotic coefficients. The expansions are simple and useful, and we discuss their use in some enumerating sequences in trees, lattice paths and planar maps.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
Keywords
  • asymptotic expansions
  • Lagrangean forms
  • saddle-point method
  • singularity analysis
  • maps

Metrics

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

References

  1. C. Banderier, P. Flajolet, G. Schaeffer, and M. Soria. Random maps, coalescing saddles, singularity analysis, and airy phenomena. Random Struct. Algorithms, 19(3-4):194-246, 2001. URL: http://dx.doi.org/10.1002/rsa.10021.
  2. E. A Bender and L. B. Richmond. A survey of the asymptotic behaviour of maps. Journal of Combinatorial Theory, Series B, 40(3):297-329, 1986. URL: http://dx.doi.org/10.1016/0095-8956(86)90086-9.
  3. O. Bernardi and E. Fusy. A bijection for triangulations, quadrangulations, pentagulations, etc. Journal of Combinatorial Theory, Series A, 119(1):218-244, 2012. URL: http://dx.doi.org/10.1016/j.jcta.2011.08.006.
  4. M. Bousquet-Mélou and G. Schaeffer. Enumeration of planar constellations. Advances in Applied Mathematics, 24(4):337-368, 2000. URL: http://dx.doi.org/10.1006/aama.1999.0673.
  5. F. Chapoton, F. Hivert, and J.-C. Novelli. A set-operad of formal fractions and dendriform-like sub-operads. Journal of Algebra, 465:322-355, 2016. URL: http://dx.doi.org/10.1016/j.jalgebra.2016.07.001.
  6. V. De Angelis. Asymptotic expansions and positivity of coefficients for large powers of analytic functions. Int. J. Math. Math. Sci., 2003(16):1003-1025, 2003. URL: http://dx.doi.org/10.1155/S0161171203205056.
  7. M. Drmota. A Bivariate Asymptotic Expansion of Coefficients of Powers of Generating Functions. Europ. J. Combinatorics, 15:139-152, 1994. Google Scholar
  8. P. Flajolet and A. M. Odlyzko. Singularity analysis of generating functions. SIAM J. Discrete Math., 3(2):216-240, 1990. URL: http://dx.doi.org/10.1137/0403019.
  9. P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, New York, NY, USA, 1 edition, 2009. Google Scholar
  10. Zh. Gao and N. C. Wormald. Enumeration of rooted cubic planar maps. Annals of Combinatorics, 6(3):313-325, Dec 2002. URL: http://dx.doi.org/10.1007/s000260200006.
  11. V. A. Liskovets and T. R. Walsh. Counting unrooted maps on the plane. Adv. in Appl. Math., 36(4):364-387, 2006. URL: http://dx.doi.org/10.1016/j.aam.2005.03.006.
  12. Y. Liu. Enumerative theory of maps, volume 468 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht; Science Press Beijing, Beijing, 1999. Google Scholar
  13. A. Meir and J. W Moon. On the altitude of nodes in random trees. Canadian Journal of Mathematics, 30(1978):997-1015, 1978. URL: http://dx.doi.org/10.4153/CJM-1978-085-0.
  14. A. Meir and J. W. Moon. The asymptotic behaviour of coefficients of powers of certain generating functions. European J. Combin., 11(6):581-587, 1990. URL: http://dx.doi.org/10.1016/S0195-6698(13)80043-1.
  15. R. C. Mullin, L. B. Richmond, and R. G. Stanton. An asymptotic relation for bicubic maps. In Proceedings of the Third Manitoba Conference on Numerical Mathematics (Winnipeg, Man., 1973), pages 345-355. Utilitas Math., Winnipeg, Man., 1974. Google Scholar
  16. R. Sprugnoli and M. C. Verri. Asymptotics for lagrange inversion. Pure Mathematics and Applications, 5(1):79-104, 1994. URL: http://EconPapers.repec.org/RePEc:cmt:pumath:puma1994v005pp0079-0104.
  17. F. G. Tricomi and A. Erdélyi. The asymptotic expansion of a ratio of gamma functions. Pacific J. Math., 1:133-142, 1951. URL: http://projecteuclid.org/euclid.pjm/1102613160.
  18. W. T. Tutte. On the enumeration of planar maps. Bull. Amer. Math. Soc., 74:64-74, 1968. URL: http://dx.doi.org/10.1090/S0002-9904-1968-11877-4.
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 Imprint Privacy Contact