The partition algebra P k (n) and the symmetric group S n are in Schur-Weyl duality on the k-fold tensor power M ⊗k n of the permutation module M n of S n , so there is a surjection, which is an isomorphism when n ≥ 2k. We prove a dimension formula for the irreducible modules of the centralizer algebra Z k (n) in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities of the irreducible S n -modules in M ⊗k n . Our dimension expressions hold for any n ≥ 1 and k ≥ 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 M ⊗k n and the quasi-partition algebra corresponding to tensor powers of the reflection representation of S n .