Let (F, R, k) be a p-modular system, and let RSn denote the centralizer of the symmetric group S in the group algebra RSn, where = n. We show that the decomposition map of RS S n can be determined from that of the degenerate affine Hecke algebra R n-of rank n -. We use this to determine the blocks of RS S n for = n -2 n -3. For each p-core, there is an n0 such that if n > n0 and En is a block idempotent of RSn with core, then EnEn- is zero or a block idempotent of RS Sn, for each block idempotent En- of RSn-.