Enumeration of partial Lukasiewicz paths

TL;DR

This paper provides generating functions, exact counts, and asymptotic analysis for partial Lukasiewicz paths, including bounded height and paths with consecutive steps, advancing combinatorial enumeration methods.

Contribution

It introduces explicit formulas and asymptotic results for counting partial Lukasiewicz paths with various constraints, including height and step patterns.

Findings

Exact enumeration formulas for path prefixes and suffixes.

Asymptotic average height of paths behaves as √πn.

Analysis of paths with consecutive steps (alternating paths).

Abstract

\L{}ukasiewicz paths are lattice paths in $\Bbb{N}^2$ starting at the origin, ending on the $x$-axis, and consisting of steps in the set $\{(1,k), k\geq -1\}$. We give generating function and exact value for the number of $n$-length prefixes (resp. suffixes) of these paths ending at height $k\geq 0$ with a given type of step. We make a similar study for prefixes of height at most $t\geq 0$. Using the explicit forms for the paths of bounded height, we evaluate the average height asymptotically. For fixed $k$ and $n\to\infty$, this quantity behaves as $\sqrt{\pi n}$. Finally we study (in the same way) prefixes of alternate \L{}ukasiewicz paths, i.e., \L{}ukasiewicz paths that do contain two consecutive steps with the same direction.