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 WagnerEmail author
Open Access


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.


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 (, 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
    Email author
  1. 1.Institut für MathematikAlpen-Adria-Universität KlagenfurtKlagenfurtAustria
  2. 2.Department of Mathematical SciencesStellenbosch UniversityStellenboschSouth Africa

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