Enumeration of colored Dyck paths via partial Bell polynomials

TL;DR

This paper introduces a method to count colored Dyck paths using partial Bell polynomials, unifying various existing formulas and enabling efficient enumeration of complex path structures.

Contribution

It presents a recurrence relation and a Bell polynomial framework for counting colored Dyck paths with diverse ascent and descent restrictions.

Findings

Derived a convolution-type recurrence relation.

Unified multiple known Dyck path formulas.

Simplified counting of colored lattice paths.

Abstract

We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block $P_j$ take on $c_j$ colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in an unified manner.