A $k$-colored partition, or $k$-multipartition is a partition in which parts may appear in $k$ different "colors", where the order of colors does not matter. For example, denoting colors by subscripts, the 2-colored partitions of 3 are $$\begin{equation*} 3_2, 3_1, 2_2+1_2, 2_2+1_1, 2_1+1_2, 2_1+1_1, 1_2+1_2+1_2, 1_2+1_2+1_1, 1_2+1_1+1_1, 1_1+1_1+1_1. \end{equation*}$$ The generating function of $c_k(n)$, the number of $k$-colored partitions of $n$, is $$\begin{equation*} C_k(q) := \sum_{n=0}^{\infty}c_k(n)q^n = \prod_{n=1}^{\infty} \frac{1}{(1 − q^n)^k}. \end{equation*}$$

William J. Keith considered restricted $k$-color partitions, by defining a $(k, j)$-colored partition of an integer $n$ to be a $k$-colored partition in which at most $j$ colors appear for any given size of part. Denote the number of $(k, j)$-colored partitions of $n$ by $c_{k,j} (n)$. Keith showed that the generating function for $c_{k,j} (n)$ is $$\begin{equation*} C_{k,j}(q) := \sum_{n=0}^{\infty}c_{k,j}(n)q^n = \prod_{n=1}^{\infty} \left (1 + \frac{\binom{k}{1}q^n}{1-q^n} + \frac{\binom{k}{2}q^{2n}}{(1-q^n)^2} + \dots + \frac{\binom{k}{j}q^{jn}}{(1-q^n)^j} \right ). \end{equation*}$$ He considered the special case $j=1$, and using known facts about overpartitions, proved a number of congruences such as $$c_{29,1}(24n + 19) \equiv 0 (\mod 783).$$ He also considered the special case $j=k-1$ and showed that the generating function is $$\begin{equation*} C_{k,k-1}(q) := \sum_{n=0}^{\infty}c_{k,k-1}(n)q^n = \prod_{n=1}^{\infty} \frac{1-q^{kn}}{(1-q^n)^k}, \end{equation*}$$ and also proved several congruences such as $$c_{5,4}(4n + 2) \equiv 0 (\mod 20).$$ Coloured parttions are also known as multipartitions, and some more information about them may be found on that page.

[1] Keith, William J. Restricted $k$-color partitions. Ramanujan J. 40 (2016), no. 1, 71–92.

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