Cyclic sieving and plethysm coefficients

Abstract

A combinatorial expression for the coefficient of the Schur function s λ s_{\lambda } in the expansion of the plethysm p n / d d ∘ s μ p_{n/d}^d \circ s_{\mu } is given for all d d dividing n n for the cases in which n = 2 n=2 or λ \lambda is rectangular. In these cases, the coefficient ⟨ p n / d d ∘ s μ , s λ ⟩ \langle p_{n/d}^d \circ s_{\mu }, s_{\lambda } \rangle is shown to count, up to sign, the number of fixed points of an ⟨ s μ n , s λ ⟩ \langle s_{\mu }^n, s_{\lambda } \rangle -element set under the d d th power of an order- n n cyclic action. If n = 2 n=2 , the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if λ \lambda 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 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.