Semisimple algebras related to immaculate tableaux

Given a direct sum $A$ of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of $A$ and the dimensions of the irreducible $A$-modules, then this can be thought of as providing an analogue of the famous Frobenius-Young identity $n!={\sum }_{\lambda ⊢n}(f^{\lambda }{)}^2$ derived from the semisimple structure of the symmetric group algebra $\mathbb{C}{S}_n$, letting $f^{\lambda }$ denote the number of Young tableaux of partition shape $\lambda ⊢n$. By letting $g^{\alpha }$ denote the number of standard immaculate tableaux of composition shape $\alpha \models n$, we construct an algebra $\mathbb{C}{\mathcal{I}}_n$ with a semisimple structure such that $\dim \mathbb{C}{\mathcal{I}}_n={\sum }_{\alpha \models n}(g^{\alpha }{)}^2$ and such that $\mathbb{C}{\mathcal{I}}_n$ contains an isomorphic copy of $\mathbb{C}{S}_n$. We bijectively prove a recurrence for $\dim \mathbb{C}{\mathcal{I}}_n$ so as to construct a basis of $\mathbb{C}{\mathcal{I}}_n$ indexed by permutation-like objects that we refer to as immacutations. We form a basis ${\mathcal{B}}_n$ of $\mathbb{C}{\mathcal{I}}_n$ such that $\mathbb{C}{\mathcal{B}}_n$ has the structure of a monoid algebra in such a way so that ${\mathcal{B}}_n$ is closed under the multiplicative operation of $\mathbb{C}{\mathcal{I}}_n$, yielding a monoid structure on the set of order-$n$ immacutations.