Motzkin paths with two variants of level steps on odd levels -- a kernel method approach

TL;DR

This paper develops a kernel method approach to enumerate Motzkin paths with two types of horizontal steps on odd levels, connecting to a known integer sequence and establishing holonomic recursions.

Contribution

It introduces a kernel method framework for counting specialized Motzkin paths with variants of level steps, including partial path enumeration and automatic recursion derivation.

Findings

Enumeration formulas for the paths are derived.

The coefficients satisfy a holonomic recursion.

The method can be adapted to other step variants.

Abstract

The sequence A176677 in the Encyclopedia of Integer Sequences enumerates Motzkin paths where two types of horizontal steps may occur, but only on odd indexed levels. We show how to perform the enumeration, also dealing with partial such Motzkin paths leading to a particular level or to any level (open paths). The method is the kernel method where functional equations are manipulated in a suitable way. The coefficients of sequence A176677 satisfy a holonomic recursion that was recently discussed on the arxiv. We show how this can be established in an (almost) automatic fashion. Eventually we switch the roles of `odd' and `even'. One could also allow more versions of horizontal steps but we leave this to the interested readers.