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We introduce the multiset partition algebra, ${\rm M\!P}_{r,k}(x)$, that hasbases elements indexed by multiset partitions, where $x$ is an indeterminateand $r$ and $k$ are non-negative integers. This algebra can be realized as adiagram algebra that generalizes the partition algebra. When $x$ is an integergreater or equal to $2r$, we show that ${\rm M\!P}_{r,k}(x)$ is isomorphic to acentralizer algebra of the symmetric group, $S_n$, acting on the polynomialring on the variables $x_{ij}$, $1\leq i \leq n$ and $1\leq j\leq k$. Wedescribe the representations of ${\rm M\!P}_{r,k}(x)$, branching rule andrestriction of its representations in the case that $x$ is an integer greateror equal to $2r$.

Howe duality of the symmetric group and a multiset partition algebra