Skew Dyck paths having no peaks at level 1

TL;DR

This paper analyzes skew Dyck paths with no peaks at level 1, extending to partial decorated paths ending at a specified level, using generating functions for a comprehensive combinatorial study.

Contribution

It introduces a detailed analysis of partial decorated Dyck paths ending at any level, expanding the understanding of skew Dyck path variants through generating functions.

Findings

Derived generating functions for skew Dyck paths with no peaks at level 1.

Extended analysis to partial paths ending at arbitrary levels.

Provided formulas for counting such paths from both directions.

Abstract

Skew Dyck paths are a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ a south-west step $(-1,-1)$ is also allowed, provided that the path does not intersect itself. Replacing the south-west step by a red south-east step, we end up with decorated Dyck paths. Sequence A128723 of the Encyclopedia of Integer Sequences considers such paths where peaks at level 1 are forbidden. We provide a thorough analysis of a more general scenario, namely partial decorated Dyck paths, ending on a prescribed level $j$, both from left-to-right and from right-to-left (decorated Dyck paths are not symmetric). The approach is completely based on generating functions.