  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.

Combinatorics