The rook partition algebra

The rook partition algebra RP k ( x ) is a generically semisimple algebra that arises from looking at what commutes with the action of the symmetric group S n on U ⊗ k , where U is the direct sum of the natural representation and the trivial representation of S n . We give a combinatorial description of this algebra, construct its irreducible representations, and exhibit a Murnaghan–Nakayama formula to compute certain character values.