Combinatorics of positional colored compositions

TL;DR

This paper studies a special class of colored compositions with location-dependent coloring rules, deriving counting sequences via generating functions and exploring their connections to various combinatorial objects.

Contribution

It introduces a novel framework for analyzing positional colored compositions, providing generating function formulas and generalizations for related combinatorial structures.

Findings

Derived explicit counting sequences using generating functions.

Established connections between positional colored compositions and other combinatorial objects.

Provided combinatorial arguments and generalizations for the observed relationships.

Abstract

We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial objects are discussed, with combinatorial arguments provided and generalized for these observations.