  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.
