Wilcox has considered a twisted semigroup algebra structure on the partition algebra \(\mathbb {C}A_k(n)\) , but it appears that there has not previously been any known basis that gives \(\mathbb {C}A_k(n)\) the structure of a “non-twisted” semigroup algebra or a monoid algebra. This motivates the following problem, for the non-degenerate case whereby \(n \in \mathbb {C} \setminus \{ 0, 1, \ldots , 2 k - 2 \}\) so that \( \mathbb {C}A_k(n)\) is semisimple. How could a basis \(M_{k} = M\) of \( \mathbb {C}A_k(n)\) be constructed so that M is closed under the multiplicative operation on \(\mathbb {C}A_k(n)\) , in such a way so that M is a monoid under this operation, and how could a product rule for elements in M be defined in an explicit and combinatorial way in terms of partition diagrams? We construct a basis M of the desired form using Halverson and Ram’s matrix unit construction for partition algebras, Benkart and Halverson’s bijection between vacillating tableaux and set-partition tableaux, an analogue given by Colmenarejo et al. for partition diagrams of the RSK correspondence, and a variant of a result due to Hewitt and Zuckerman characterizing finite-dimensional semisimple algebras that are isomorphic to semigroup algebras.
