CYCLIC SIEVING AND PLETHYSM COEFFICIENTS.

A combinatorial expression for the coefficient of the Schur function sλ in the expansion of the plethysm pn/dd ... sμ is given for all d dividing n for the cases in which n = 2 or λ is rectangular. In these cases, the coefficient <pn/dd ... sμ, sλ> is shown to count, up to sign, the number of fixed points of an <sμn, sλ>-element set under the dth power of an order-n cyclic action. If n = 2, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if λ is rectangular, the action is a certain power of Schützenberger and Shimozono's jeu-de-taquin promotion. This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case n = 2 is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. [ABSTRACT FROM AUTHOR]