Enumeration of partial Lukasiewicz paths

Łukasiewicz 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 $\sqrt{πn}$. Finally we study (in the same way) prefixes of alternate Łukasiewicz paths, i.e., Łukasiewicz paths that do contain two consecutive steps with the same direction.

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