  Schur-Weyl duality concerns the actions of $\text{GL}_{n}(\mathbb{C})$ and
$S_{k}$ on tensor powers of the form $V^{\otimes k}$ for an $n$-dimensional
vector space $V$. There are rich histories within representation theory,
combinatorics, and statistical mechanics involving the study and use of diagram
algebras, which arise through the restriction of the action of
$\text{GL}_{n}(\mathbb{C})$ to subgroups of $\text{GL}_{n}(\mathbb{C})$. This
leads us to consider further variants of Schur-Weyl duality, with the use of
variants of the tensor space $V^{\otimes k}$. Instead of taking repeated tensor
products of $V$, we make use of a freest commutative bilinear operation in
place of $\otimes$, and this is motivated by an associated invariance property
given by the action of $S_{k}$. By then taking the centralizer algebra with
respect to the action of the group of permutation matrices in
$\text{GL}_{n}(\mathbb{C})$, this gives rise to a diagram-like algebra spanned
by a new class of combinatorial objects. We construct orbit-type bases for the
centralizer algebras introduced in this paper, and we apply these bases to
prove a combinatorial formula for the dimensions of our centralizer algebras.
