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We prove a conjecture of Stanley on the-index of the semisuspension of the face poset of a simplicial shelling component. We give a new signed generalization of André permutations, together with a new notion of-variation for signed permutations. This generalization not only allows us to compute the-index of the face poset of a cube, but also occurs as a natural set of orbit representatives for a signed generalization of the Foata-Strehl commutative group action on the symmetric group. From the induction techniques used, it becomes clear that there is more than one way to define classes of permutations and-variation such that they allow us to compute the-index of the same poset.

On thecd-variation polynomials of André and simsun permutations