Ascents in Non-Negative Lattice Paths

Abstract

Non-negative {\L}ukasiewicz paths are special two-dimensional lattice paths never passing below their starting altitude which have only one single special type of down step. They are well-known and -studied combinatorial objects, in particular due to their bijective relation to trees with given node degrees. We study the asymptotic behavior of the number of ascents (i.e., the number of maximal sequences of consecutive up steps) of given length for classical subfamilies of general non-negative {\L}ukasiewicz paths: those with arbitrary ending altitude, those ending on their starting altitude, and a variation thereof. Our results include precise asymptotic expansions for the expected number of such ascents as well as for the corresponding variance.