# Polynomial analogues of restricted multicolor b-ary partition functions

## TL;DR

This paper introduces polynomial representations for restricted multicolor b-ary partition functions, providing explicit formulas, recurrence relations, and a factorization theorem to characterize these partitions.

## Contribution

It develops a novel polynomial framework for restricted multicolor b-ary partitions, including explicit formulas and recurrence relations.

## Key findings

- Polynomials characterize all restricted multicolor b-ary partitions.
- Derived explicit formulas for the polynomials.
- Established a recurrence relation and a factorization theorem.

## Abstract

Given an integer base $b\geq 2$, a number $\rho\geq 1$ of colors, and a finite sequence $\Lambda=(\lambda_1,\ldots,\lambda_\rho)$ of positive integers, we introduce the concept of a $\Lambda$-restricted $\rho$-colored $b$-ary partition of an integer $n\geq 1$. We also define a sequence of polynomials in $\lambda_1+\cdots+\lambda_\rho$ variables, and prove that the $n$th polynomial characterizes all $\Lambda$-restricted $\rho$-colored $b$-ary partitions of $n$. In the process we define a recurrence relation for the polynomials in question, obtain explicit formulas and identify a factorization theorem.