A Symmetry Theorem on a Modified Jeu De Taquin

For their bijective proof of the hook-length formula for the number of standard tableaux of a fixed shape Novelli et al. define a modified jeu de taquin which transforms an arbitrary filling of the Ferrers diagram with 1, 2,⋯ , n(tabloid) into a standard tableau. Their definition relies on a total order of the cells in the Ferrers diagram induced by a special standard tableau, however, this definition also makes sense for the total order induced by any other standard tableau. Given two standard tableaux P, Q of the same shape we show that the number of tabloids which result in P if we perform the modified jeu de taquin with respect to the total order induced by Q is equal to the number of tabloids which result in Q if we perform the modified jeu de taquin with respect to P. This symmetry theorem extends to skew shapes and shifted skew shapes.