The rook partition algebra

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