Ascents in Non-Negative Lattice Paths

TL;DR

This paper analyzes the asymptotic behavior of the number of ascents in non-negative Lukasiewicz paths, providing precise formulas for their expected value and variance across various path subfamilies.

Contribution

It offers the first detailed asymptotic analysis of ascent counts in non-negative Lukasiewicz paths, including explicit formulas for expectation and variance.

Findings

Derived asymptotic expansions for expected ascents.

Calculated variance of ascent counts.

Analyzed different path subfamilies and their ascent behaviors.

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.