License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2020.2
URN: urn:nbn:de:0030-drops-120329
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12032/
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Banderier, Cyril ; Lackner, Marie-Louise ; Wallner, Michael

Latticepathology and Symmetric Functions (Extended Abstract)

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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).

BibTeX - Entry

@InProceedings{banderier_et_al:LIPIcs:2020:12032,
  author =	{Cyril Banderier and Marie-Louise Lackner and Michael Wallner},
  title =	{{Latticepathology and Symmetric Functions (Extended Abstract)}},
  booktitle =	{31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)},
  pages =	{2:1--2:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-147-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{159},
  editor =	{Michael Drmota and Clemens Heuberger},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12032},
  URN =		{urn:nbn:de:0030-drops-120329},
  doi =		{10.4230/LIPIcs.AofA.2020.2},
  annote =	{Keywords: Lattice path, generating function, symmetric function, algebraic function, kernel method, context-free grammar, Sparre Andersen formula, Spitzer’s identity, Wiener - Hopf factorization}
}

Keywords: Lattice path, generating function, symmetric function, algebraic function, kernel method, context-free grammar, Sparre Andersen formula, Spitzer’s identity, Wiener - Hopf factorization
Collection: 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)
Issue Date: 2020
Date of publication: 10.06.2020

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