Annals of Combinatorics

, Volume 20, Issue 4, pp 775–797 | Cite as

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

  • Benjamin Hackl
  • Clemens Heuberger
  • Helmut Prodinger
  • Stephan Wagner
Open Access
Article

Abstract

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.

Keywords

lattice path culminating path ballot sequence asymptotic expansion Chebyshev polynomial 

Mathematics Subject Classification

05A16 05A15 05A10 60C05 

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Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Benjamin Hackl
    • 1
  • Clemens Heuberger
    • 1
  • Helmut Prodinger
    • 2
  • Stephan Wagner
    • 2
  1. 1.Institut für MathematikAlpen-Adria-Universität KlagenfurtKlagenfurtAustria
  2. 2.Department of Mathematical SciencesStellenbosch UniversityStellenboschSouth Africa

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