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.
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B. Hackl and C. Heuberger are supported by the Austrian Science Fund (FWF): P 24644-N26.
H. Prodinger is supported by an incentive grant of the National Research Foundation of South Africa.
S. Wagner is supported by the National Research Foundation of South Africa, grant number 96236.
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Hackl, B., Heuberger, C., Prodinger, H. et al. Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum. Ann. Comb. 20, 775–797 (2016). https://doi.org/10.1007/s00026-016-0330-0
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DOI: https://doi.org/10.1007/s00026-016-0330-0