Weighted enumeration of even-k Motzkin paths

Compute the weighted enumeration of even-k Motzkin paths, for arbitrary nonnegative size n and suitable weight functions f from the positive integers to the complex numbers, and/or derive a functional equation satisfied by the corresponding generating function.

Background

The paper defines k-Motzkin paths using different step sets according to the parity of k. For even k, these paths use steps with vertical increments 2k, k-1, and -2. The general weighted-enumeration framework developed earlier in the paper requires down-steps of vertical size 1, so it applies to odd-k Motzkin paths but not to the even-k case because of the (1,-2) step.

The authors use the kernel method to obtain an explicit formula for the unweighted number of even-k Motzkin paths, but leave the corresponding enumeration with ascent-block weights unresolved. The open problem asks for such a weighted enumeration and/or a functional equation for its generating function.

References

Following the discussions in Section \ref{sec:motzkin}, a natural open question is the weighted enumeration of $k$-Motzkin paths when $k$ is even. For $n\in\mathbb{Z}{\geq0}$, even $k\in\mathbb{Z}+$, and certain weight function $f:\mathbb{Z}_+\to\mathbb{C}$, compute

\sum_{M\in\mathcal{M}n{(k)}\prod{i=1}{|\mathbf{u}(M)|}f(\mathbf{u}(M)_i),

and/or find a function equation that its generating function satisfies.

— Lattice paths enumerations weighted by ascent lengths  (2501.01152 - Yan, 2 Jan 2025) in Section 6, Concluding remarks