The symmetric group $\mathsf{ S}_n$ and the partition algebra $\mathsf{ P}_k(n)$ centralize one another in their actions on the $k$-fold tensor power $\mathsf{ M}_n^{ \otimes k}$ of the $n$-dimensional permutation module $\mathsf{ M}_n$ of $\mathsf{ S}_n$. The duality afforded by the commuting actions determines an algebra homomorphism $\Phi_{ k,n}: \mathsf{ P}_k(n) \to \mathsf{ End}_{ \mathsf{ S}_n}(\mathsf{ M}_n^{ \otimes k})$ from the partition algebra to the centralizer algebra $\mathsf{ End}_{ \mathsf{ S}_n}(\mathsf{ M}_n^{ \otimes k})$, which is a surjection for all $k, n \in \mathbb{ Z}_{ \ge 1}$, and an isomorphism when $n \ge 2k$. We present results that can be derived from the duality between $\mathsf{ S}_n$ and $\mathsf{ P}_k(n)$; for example, (i) expressions for the multiplicities of the irreducible $\mathsf{ S}_n$-summands of $\mathsf{ M}_n^{ \otimes k}$, (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra $\mathsf{ End}_{ \mathsf{ S}_n}(\mathsf{ M}_n^{ \otimes k})$, (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra $\mathsf{ P}_k(n)$. When $2k >n$, the map $\Phi_{ k,n}$ has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of $\Phi_{ k,n}$ in terms of the orbit basis of $\mathsf{ P}_k(n)$ and explain how the surjection $\Phi_{ k,n}$ can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.
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