The partition algebra $\mathsf{ P}_k(n)$ and the symmetric group $\mathsf{ S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{ M}_n^{ \otimes k}$ of the permutation module $\mathsf{ M}_n$ of $\mathsf{ S}_n$, so there is a surjection $\mathsf{ P}_k(n) \to \mathsf{ Z}_k(n) := \mathsf{ End}_{ \mathsf{ S}_n}(\mathsf{ M}_n^{ \otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{ Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities of the irreducible $\mathsf{ S}_n$-modules in $\mathsf{ M}_n^{ \otimes k}$. Our dimension expressions hold for any $n \geq 1$ and $k\ge0$. Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on $\mathsf{ M}_n^{ \otimes k}$ and the quasi-partition algebra corresponding to tensor powers of the reflection representation of $\mathsf{ S}_n$.
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