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Latticepathology and Symmetric Functions (Extended Abstract)

Authors Cyril Banderier , Marie-Louise Lackner , Michael Wallner



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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

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 AsGet 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

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