Document Open Access Logo

Latticepathology and Symmetric Functions (Extended Abstract)

Authors Cyril Banderier , Marie-Louise Lackner , Michael Wallner

PDF
Thumbnail PDF

File

LIPIcs.AofA.2020.2.pdf
  • Filesize: 0.59 MB
  • 16 pages

Document Identifiers

Author Details

Cyril Banderier
  • Université Paris 13, LIPN, UMR CNRS 7030, France
Marie-Louise Lackner
  • Christian Doppler Laboratory for Artificial Intelligence and Optimization for Planning and Scheduling, DBAI, TU Wien, Austria
Michael Wallner
  • Université de Bordeaux, LaBRI, UMR CNRS 5800, France
  • Institute for Discrete Mathematics and Geometry, TU Wien, Austria

Funding

This work was initiated during the sojourn of Marie-Louise Lackner and Michael Wallner at the University of Paris Nord, via a Franco-Austrian "Amadeus" project and via a MathStic funding. The subsequent part of this collaboration was funded by the Erwin Schrödinger Fellowship of the Austrian Science Fund (FWF): J 4162-N35.

Acknowledgements

We thank our referees for their careful reading. In this period of worldwide lockdown due to the COVID19 pandemic, let us also thank Klaus Hulek, Barbara Strazzabosco, and the staff from zbMATH who implemented some technical solution so that we can have home access to this wonderful database.

Cite As Get BibTex

Cyril Banderier, Marie-Louise Lackner, and Michael Wallner. Latticepathology and Symmetric Functions (Extended Abstract). In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.AofA.2020.2

Abstract

In this article, we revisit and extend a list of formulas based on lattice path surgery: cut-and-paste methods, factorizations, the kernel method, etc. For this purpose, we focus on the natural model of directed lattice paths (also called generalized Dyck paths). We introduce the notion of prime walks, which appear to be the key structure to get natural decompositions of excursions, meanders, bridges, directly leading to the associated context-free grammars. This allows us to give bijective proofs of bivariate versions of Spitzer/Sparre Andersen/Wiener - Hopf formulas, thus capturing joint distributions. We also show that each of the fundamental families of symmetric polynomials corresponds to a lattice path generating function, and that these symmetric polynomials are accordingly needed to express the asymptotic enumeration of these paths and some parameters of limit laws. En passant, we give two other small results which have their own interest for folklore conjectures of lattice paths (non-analyticity of the small roots in the kernel method, and universal positivity of the variability condition occurring in many Gaussian limit law schemes).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Distribution functions
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Grammars and context-free languages
Keywords
  • Lattice path
  • generating function
  • symmetric function
  • algebraic function
  • kernel method
  • context-free grammar
  • Sparre Andersen formula
  • Spitzer’s identity
  • Wiener - Hopf factorization

Metrics

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

References

  1. David Aldous, Grégory Miermont, and Jim Pitman. The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity. Probab. Theory Related Fields, 129(2):182-218, 2004. URL: https://doi.org/10.1007/s00440-003-0334-7.
  2. Erik Sparre Andersen. On the number of positive sums of random variables. Scandinavian Actuarial Journal, 1949(1):27-36, 1949. URL: https://doi.org/10.1080/03461238.1949.10419756.
  3. Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger. Analytic combinatorics of lattice paths with forbidden patterns: enumerative aspects. In LATA'18, volume 10792 of Lecture Notes in Comput. Sci., pages 195-206. Springer, 2018. URL: https://lipn.univ-paris13.fr/~banderier/Papers/lata2018.pdf.
  4. Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, 281(1-2):37-80, 2002. URL: https://doi.org/10.1016/S0304-3975(02)00007-5.
  5. Cyril Banderier and Pierre Nicodème. Bounded discrete walks. Discrete Math. Theor. Comput. Sci., AM:35-48, 2010. URL: https://dmtcs.episciences.org/2792/pdf.
  6. Cyril Banderier and Michael Wallner. The kernel method for lattice paths below a rational slope. In Lattice paths combinatorics and applications, Developments in Mathematics Series, pages 119-154. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11102-1_7.
  7. Jean Bertoin and Jim Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math., 118(2):147-166, 1994. URL: https://digitalassets.lib.berkeley.edu/sdtr/ucb/text/350.pdf.
  8. Olivier Bodini and Yann Ponty. Multi-dimensional Boltzmann sampling of languages. Discrete Math. Theor. Comput. Sci. Proc., AM:49-63, 2010. URL: https://hal.inria.fr/hal-00450763v4/document.
  9. Mireille Bousquet-Mélou. Discrete excursions. Sém. Lothar. Combin., 57:23 pp., 2008. URL: https://www.mat.univie.ac.at/~slc/wpapers/s57bousquet.pdf.
  10. Mireille Bousquet-Mélou and Marko Petkovšek. Linear recurrences with constant coefficients: the multivariate case. Discrete Mathematics, 225(1-3):51-75, 2000. URL: https://doi.org/10.1016/S0012-365X(00)00147-3.
  11. Mireille Bousquet-Mélou and Yann Ponty. Culminating paths. Discrete Math. Theor. Comput. Sci., 10(2):125-152, 2008. URL: https://hal.archives-ouvertes.fr/hal-00151979v2/document.
  12. Alan J. Bray, Satya N. Majumdar, and Grégory Schehr. Persistence and first-passage properties in nonequilibrium systems. Advances in Physics, 62(3):225-361, 2013. URL: https://doi.org/10.1080/00018732.2013.803819.
  13. Philippe Chassaing and Svante Janson. A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Ann. Probab., 29(4):1755-1779, 2001. URL: https://doi.org/10.1214/aop/1015345771.
  14. Philippe Di Francesco and Rinat Kedem. Discrete non-commutative integrability: Proof of a conjecture by M. Kontsevich. International Mathematics Research Notices, 2010(21):4042-4063, 2010. URL: https://doi.org/10.1093/imrn/rnq024.
  15. Catherine Donati-Martin, Hiroyuki Matsumoto, and Marc Yor. On striking identities about the exponential functionals of the Brownian bridge and Brownian motion. Period. Math. Hungar., 41(1-2):103-119, 2000. URL: https://doi.org/10.1023/A:1010308203346.
  16. Philippe Duchon. On the enumeration and generation of generalized Dyck words. Discrete Mathematics, 225(1-3):121-135, 2000. URL: https://doi.org/10.1016/S0012-365X(00)00150-3.
  17. Philippe Duchon, Philippe Flajolet, Guy Louchard, and Gilles Schaeffer. Boltzmann samplers for the random generation of combinatorial structures. Combinatorics, Probability and Computing, 13(4-5):577-625, 2004. URL: https://doi.org/10.1017/S0963548304006315.
  18. Kurusch Ebrahimi-Fard, Li Guo, and Dirk Kreimer. Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT. J. Phys. A, 37(45):11037-11052, 2004. URL: https://doi.org/10.1088/0305-4470/37/45/020.
  19. William Feller. An Introduction to Probability Theory and its Applications. Vol. II. Second edition. John Wiley & Sons, 1971. Google Scholar
  20. Philippe Flajolet. Combinatorial aspects of continued fractions. Discrete Mathematics, 32(2):125-161, 1980. URL: https://doi.org/10.1016/0012-365X(80)90050-3.
  21. Philippe Flajolet and Andrew M. Odlyzko. Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumerations. Math. Proc. Cambridge Philos. Soc., 96(2):237-253, 1984. URL: https://doi.org/10.1017/S0305004100062149.
  22. Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. URL: http://algo.inria.fr/flajolet/Publications/book.pdf.
  23. Sergey Fomin and Curtis Greene. Noncommutative Schur functions and their applications. Discrete Math., 193(1-3):179-200, 1998. URL: https://doi.org/10.1016/S0012-365X(98)00140-X.
  24. Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh, and Jean-Yves Thibon. Noncommutative symmetric functions. Adv. Math., 112(2):218-348, 1995. URL: https://doi.org/10.1006/aima.1995.1032.
  25. Ira M. Gessel. A factorization for formal Laurent series and lattice path enumeration. Journal of Combinatorial Theory, Series A, 28(3):321-337, 1980. URL: https://doi.org/10.1016/0097-3165(80)90074-6.
  26. Priscilla Greenwood. Wiener-Hopf methods, decompositions, and factorisation identities for maxima and minima of homogeneous random processes. Advances in Appl. Probability, 7(4):767-785, 1975. URL: https://doi.org/10.2307/1426398.
  27. John E. Hopcroft, Raghavan Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 3rd edition, 2006. Google Scholar
  28. Thierry Jeulin and Marc Yor. Grossissements de filtrations: exemples et applications. In Séminaire de Calcul Stochastique 1982/83, Université Paris VI, volume 1118 of Lecture Notes in Math. Springer, 1985. URL: https://doi.org/10.1007/BFb0075765.
  29. Maxim Kontsevich. Noncommutative identities. arXiv, pages 1-10, 2011. URL: http://arxiv.org/abs/1109.2469v1.
  30. Christian Krattenthaler. Lattice Path Enumeration. CRC Press, 2015. In: Handbook of Enumerative Combinatorics, M. Bóna (ed.), Discrete Math. and Its Appl. URL: http://arxiv.org/abs/1503.05930.
  31. Alexey Kuznetsov, Andreas E. Kyprianou, and Juan C. Pardo. Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab., 22(3):1101-1135, 2012. URL: https://doi.org/10.1214/11-AAP787.
  32. Jacques Labelle and Yeong N. Yeh. Generalized Dyck paths. Discrete Mathematics, 82(1):1-6, 1990. URL: https://doi.org/10.1016/0012-365X(90)90039-K.
  33. Titus Lupu, Jim Pitman, and Wenpin Tang. The Vervaat transform of Brownian bridges and Brownian motion. Electron. J. Probab., 20:no. 51, 31, 2015. URL: https://doi.org/10.1214/EJP.v20-3744.
  34. Philippe Marchal. On a new Wiener-Hopf factorization by Alili and Doney. In Sém. de Probabilités, XXXV, Lecture Notes in Math. #1755, pages 416-420. Springer, 2001. URL: https://doi.org/10.1007/978-3-540-44671-2_28.
  35. Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and Cecilia Verri. Underdiagonal lattice paths with unrestricted steps. Discrete Appl. Math., 91(1-3):197-213, 1999. URL: https://doi.org/10.1016/S0166-218X(98)00126-7.
  36. Christophe Reutenauer and Marco Robado. On an algebraicity theorem of Kontsevich. Discrete Math. Theor. Comput. Sci., AR:239-246, 2012. 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012). URL: https://hal.archives-ouvertes.fr/hal-01283155/document.
  37. Frank Spitzer. A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc., 82:323-339, 1956. URL: https://doi.org/10.2307/1993051.
  38. Richard P. Stanley. Enumerative Combinatorics. Vol. 2, volume 62. Cambridge University Press, 1999. URL: https://doi.org/10.1017/CBO9780511609589.
  39. J. Michael Steele. The Bohnenblust-Spitzer algorithm and its applications. J. Comput. Appl. Math., 142(1):235-249, 2002. URL: https://doi.org/10.1016/S0377-0427(01)00472-1.
  40. Wim Vervaat. A relation between Brownian bridge and Brownian excursion. Ann. Probab., 7(1):143-149, 1979. URL: https://www.jstor.org/stable/2242845.
  41. Michael Wallner. A half-normal distribution scheme for generating functions. European J. Combin., 87(103138):1-21, 2020. http://arxiv.org/abs/1610.00541, URL: https://doi.org/10.1016/j.ejc.2020.103138.
  42. James G. Wendel. Spitzer’s formula: a short proof. Proc. Amer. Math. Soc., 9:905-908, 1958. URL: https://doi.org/10.2307/2033326.
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