Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum
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.
Keywordslattice path culminating path ballot sequence asymptotic expansion Chebyshev polynomial
Mathematics Subject Classification05A16 05A15 05A10 60C05
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