$(an+b)$-color compositions

TL;DR

This paper studies a generalized class of compositions where parts are colored based on a linear function, providing enumeration formulas via algebraic and combinatorial methods, and explores special cases with negative parameters.

Contribution

It introduces a new framework for $(an+b)$-color compositions, deriving explicit formulas using Bell polynomials and bijections, and extends the interpretation to negative parameter cases.

Findings

Derived a formula for counting $(an+b)$-color compositions with $k$ parts.

Established a bijection to domino compositions for enumeration.

Explored combinatorial interpretations for negative $b$ cases.

Abstract

For $a,b\in\mathbb{N}_0$, we consider $(an+b)$-color compositions of a positive integer $\nu$ for which each part of size $n$ admits $an+b$ colors. We study these compositions from the enumerative point of view and give a formula for the number of $(an+b)$-color compositions of $\nu$ with $k$ parts. Our formula is obtained in two different ways: 1) by means of algebraic properties of partial Bell polynomials, and 2) through a bijection to a certain family of weak compositions that we call domino compositions. We also discuss two cases when $b$ is negative and give corresponding combinatorial interpretations.