Partial Dyck path interpretation for three sequences in the Encyclopedia of Integer Sequences

TL;DR

This paper explores the combinatorial properties of Dyck paths, focusing on descents of odd length, using generating functions and the kernel method to analyze sequences from the Encyclopedia of Integer Sequences.

Contribution

It introduces a novel interpretation of partial Dyck paths for three specific sequences, extending existing relations from the Encyclopedia with new combinatorial insights.

Findings

Derived new formulas for descents of odd length in Dyck paths

Connected Dyck path properties to three integer sequences

Enhanced understanding of Dyck path variations and their generating functions

Abstract

Descents of odd length in Dyck paths are discussed, taking care of some variations. The approach is based on generating functions and the kernel method and augments relations about them from the Encyclopedia of Integer Sequences, that were pointed out by David Callan.