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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