Compositions colored by simplicial polytopic numbers

TL;DR

This paper introduces a new class of colored compositions based on simplicial polytopic numbers, providing explicit enumeration formulas and bijections to other composition sets, generalizing previous color composition results.

Contribution

It generalizes existing results for color compositions by introducing compositions colored according to simplicial polytopic numbers and provides explicit formulas and bijections.

Findings

Derived explicit formulas for these generalized compositions.

Established bijections to compositions with restricted part sizes.

Extended known results from classical color compositions to a broader class.

Abstract

For a given integer $d\ge 1$, we consider $\binom{n+d-1}{d}$-color compositions of a positive integer $\nu$ for which each part of size $n$ admits $\binom{n+d-1}{d}$ colors. We give explicit formulas for the enumeration of such compositions, generalizing existing results for $n$-color compositions (case $d=1$) and $\binom{n+1}{2}$-color compositions (case $d=2$). In addition, we give bijections from the set of $\binom{n+d-1}{d}$-color compositions of $\nu$ to the set of compositions of $(d+1)\nu - 1$ having only parts of size $1$ and $d+1$, the set of compositions of $(d+1)\nu$ having only parts of size congruent to $1$ modulo $d+1$, and the set of compositions of $(d+1)\nu + d$ having no parts of size less than $d+1$. Our results rely on basic properties of partial Bell polynomials and on a suitable adaptation of known bijections for $n$-color compositions.