A multipartition of a positive integer $n$ with $k$ components is a $k$-tuple \[ \lambda = (\lambda (1) ,\dots ,\lambda (k) ) \] of partitions such that \[ |\lambda (1) | + ··· + |\lambda (k) | = n. \] If $k$ is understood, this is more simply referred to as a multipartition of $n$. Note that some of the components in a multipartition may be empty. As an example, the multipartitions of 3 into two components may be listed as \begin{multline*} (3,-),\quad (-,3),\quad (2+1,-),\quad (2,1),\quad (1,2),\quad (-,2+1),\\ (1+1+1,-),\quad (1+1,1),\quad (1,1+1),\quad (-,1+1+1). \end{multline*} A multipartition with $k$ components is also known as a $k$-colored partition,, and more information about them may be found on that page.

If $P_k(n)$ denotes the number of multipartions of $n$, then the generating function for the sequence $\{P_k(n)\}_{n=0}^{\infty}$ has closed form given by \begin{equation}\label{multiparteq1} \sum_{n=0}^{\infty}P_k(n)q^n =\prod_{j=1}^{\infty}\frac{1}{(1-q^{j})^k}. \end{equation}

As well as being objects of interest in their own right, multipartitions also have connections with other areas of mathematics, such as Lie algebras. A survey of some of their properties are found in the paper [1] of Andrews. One of the results he proved in this survey paper was the following theorem.

Theorem. For every prime $p > 3$, there are $(p+1)/2$ values of $b$ in the interval $[1,p]$ for which \[ P_{p−3} (pn + b)\equiv 0 (\mod p) \] for all $n\geq 0$.
For example, for $p = 5$, the values of $b$ are 2, 3, 4; for $p = 7$, they are 2, 4, 5, 6, and for $p = 11$ the values of $b$ are 2, 4, 5, 7, 8, 9.

[1] George E. Andrews, A survey of multipartitions: congruences and identities. Surveys in number theory, 1–19, Dev. Math., 17, Springer, New York, 2008.

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