Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

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 $$\mathsf{P}$_k(n) \to $\mathsf{Z}$k(n) := $\mathsf{E}\mathsf{n}\mathsf{d}${$\mathsf{S}$_n}($\mathsf{M}$_n^{\otimes k}),$ which is an isomorphism when $n\ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra ${\mathsf{Z}}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities of the irreducible ${\mathsf{S}}_n$-modules in ${\mathsf{M}}_n^{\otimes k}$. Our dimension expressions hold for any $n\ge 1$ and $k\ge 0$. Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on $$\mathsf{M}$_n^{\otimes k}$ and the quasi-partition algebra corresponding to tensor powers of the reflection representation of ${\mathsf{S}}_n$.