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.

The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric group, it is perhaps not surprising that the construction outlined in this paper is very similar to the classic combinatorial construction of the irreducible $S_n$-representations: namely, the Specht modules.

Abstract. Associated with a simple graph G is a family of real, symmetric zero diagonal 1 matrices with the same nonzero pattern as the adjacency matrix of G. The minimum of the ranks of 2 the matrices in this family is denoted mr 0 (G). We characterize all connected graphs G with extreme 3 minimum zero-diagonal rank: a connected graph G has mr 0 (G) ≤ 3 if and only if it is a complete 4 multipartite graph, and mr 0 (G) = |G| if and only if it has a unique spanning generalized cycle (also 5 called a perfect [1, 2]-factor). We present an algorithm for determining whether a graph has a unique 6 spanning generalized cycle. In addition, we determine maximum zero-diagonal rank and show that 7 for some graphs, not all ranks between minimum and maximum zero-diagonal ranks are allowed.

In 1937, Richard Brauer identified the centralizer algebra of transformations Ž . commuting with the action of the complex special orthogonal groups SO 2 n .Ž . Ž mk . nCorresponding to the centralizer algebra E 2 n s End

The study of token addition and removal and token jumping reconfiguration graphs for power domination is initiated. Some results established here can be extended by applying the methods used for power domination to reconfiguration graphs for other parameters such as domination and zero forcing, so these results are first established in a universal framework.