We introduce the multiset partition algebra MPk(ξ)\documentclass[12pt]{minimal}
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\begin{document}$${\cal M}{{\cal P}_k}\left(\xi \right)$$\end{document} over the polynomial ring F[ξ], where F is a field of characteristic 0 and k is a positive integer. When ξ is specialized to a positive integer n, we establish the Schur—Weyl duality between the actions of resulting algebra MPk(n)\documentclass[12pt]{minimal}
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\begin{document}$${\cal M}{{\cal P}_k}\left(n \right)$$\end{document} and the symmetric group Sn on Symk(Fn). The construction of MPk(ξ)\documentclass[12pt]{minimal}
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\begin{document}$${\cal M}{{\cal P}_k}\left(\xi \right)$$\end{document} generalizes to any vector λ of non-negative integers yielding the algebra MPλ(ξ)\documentclass[12pt]{minimal}
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\begin{document}$${\cal M}{{\cal P}_\lambda}\left(\xi \right)$$\end{document} over F[ξ] so that there is Schur—Weyl duality between the actions of MPλ(n)\documentclass[12pt]{minimal}
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\begin{document}$${\cal M}{{\cal P}_\lambda}\left(n \right)$$\end{document} and Sn on Symλ(Fn). We find the generating function for the multiplicity of each irreducible representation of Sn in Symλ(Fn), as λ varies, in terms of a plethysm of Schur functions. As consequences we obtain an indexing set for the irreducible representations of MPk(n)\documentclass[12pt]{minimal}
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\begin{document}$${\cal M}{{\cal P}_k}\left(n \right)$$\end{document} and the generating function for the multiplicity of an irreducible polynomial representation of GLn(F) when restricted to Sn. We show that MPλ(ξ)\documentclass[12pt]{minimal}
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\begin{document}$${\cal M}{{\cal P}_\lambda}\left(\xi \right)$$\end{document} embeds inside the partition algebra P|λ|(ξ)\documentclass[12pt]{minimal}
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\begin{document}$${{\cal P}_{\left| \lambda \right|}}\left(\xi \right)$$\end{document}. Using this embedding, we show that the multiset partition algebras are generically semisimple over F. Also, for the specialization of ξ at v in F, we prove that MPλ(v)\documentclass[12pt]{minimal}
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\begin{document}$${\cal M}{{\cal P}_\lambda}\left(v \right)$$\end{document} is a cellular algebra.
