2015/03/31 by Benjamin Hackl, Clemens Heuberger, Helmut Prodinger +1
Mathematics · #Advanced Combinatorial Mathematics #Ballot #Bijection #Conjecture #Lattice (music) #Path (computing) #Random Matrices and Applications #Random walk #Stochastic processes and statistical mechanics #math.CO #msc:05A10 #msc:05A15 #msc:05A16 #msc:60C05
paper · pdf · doi:10.1007/s00026-016-0330-0
published as Ann. Comb. (2016) 20: 775 - 797
arxiv created 2015/10/12 · openalex created_date 2016/06/24 · openalex publication_date 2016/11/01 · arxiv updated 2016/11/16 · openalex updated_date 2026/08/06
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.