On Enumeration of Dyck--Schr\"oder Paths

TL;DR

This paper develops explicit generating functions for enumerating Dyck--Schr"oder paths with specific step constraints, providing efficient computational methods and connecting to known integer sequences.

Contribution

It introduces new enumeration formulas for Dyck--Schr"oder paths with different step restrictions, enhancing understanding and computational tools for these combinatorial objects.

Findings

Derived explicit generating functions for path enumeration

Provided efficient algorithms for sequence computation

Connected enumeration results to OEIS sequences

Abstract

We address the problem of enumerating paths in square lattices, where allowed steps include (1,0) and (0,1) everywhere, and (1,1) above the diagonal y=x. We consider two such lattices differing in whether the (1,1) steps are allowed along the diagonal itself. Our analysis leads to explicit generating functions and an efficient way to compute terms of many sequences in the Online Encyclopedia of Integer Sequences, proposed by Clark Kimberling almost two decades ago.