$(an+b)$-color compositions
Daniel Birmajer, Juan B. Gil, Michael D. Weiner

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 , we consider -color compositions of a positive integer for which each part of size admits colors. We study these compositions from the enumerative point of view and give a formula for the number of -color compositions of with 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 is negative and give corresponding combinatorial interpretations.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · semigroups and automata theory
