Fixed points of the evacuation of maximal chains on fuss shapes

For a partition λ of an integer, we associate λ with a slender poset P the Hasse diagram of which resembles the Ferrers diagram of λ. Let X be the set of maximal chains of P. We consider Stanley’s involution ɛ: X → X, which is extended from Schützenberger’s evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map ɛ: X → X when λ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge’s q = −1 phenomenon for the maximal chains of P under the involution ɛ for the restricted shapes.