On Enumeration of Dyck Paths with colored hills

TL;DR

This paper explores properties of functions related to Dyck paths with colored hills, providing explicit formulas, combinatorial interpretations, and identities, especially focusing on functions derived from Fine numbers and their relation to Catalan triangles.

Contribution

It introduces new explicit formulas and combinatorial interpretations for functions based on Fine numbers, connecting them to well-known structures like Catalan triangles.

Findings

g2 and g3 are Catalan triangles

Derived explicit formulas for functions f_i and g_i

Established a ten-item combinatorial identity

Abstract

We continue to investigate the properties of the earlier defined functions fm and gm, which depend on an initial arithmetic function f0. In this papers values of f0 are the Fine numbers. We investigate functions fi; gi; (i = 1; 2; 3; 4). For each function, we derive an explicit formula and give a combinatorial interpretation. It appears that g2 and g3 are well-known combinatoric object called the Catalan triangles. We finish with an identity consisting of ten items.