Domino tableaux, Schützenberger involution, and the symmetric group action

We define an action of the symmetric group S[ n 2 ] on the set of domino tableaux, and prove that the number of domino tableaux of weight β does not depend on the permutation of the weight β. A bijective proof of the well-known result due to J. Stembridge that the number of self–evacuating tableaux of a given shape and weight β = (β1, . . . , β[ n+1 2 ], β[ n2 ], . . . , β1), is equal to that of domino tableaux of the same shape and weight β = (β1, . . . , β[ n+1 2 ]) is given.