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

The partition algebra P-k (n) and the symmetric group S-n are in Schur-Weyl duality on the k-fold tensor power M-n(circle times k) of the permutation module M-n of S-n, so there is a surjection P-k (n) -> Z(k) (n) := End(Sn) (M-n(circle times k)), 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-n(circle times k). 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-n(circle times k) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of S-n.