Fixed points of the evacuation of maximal chains on fuss shapes (Q1700802)

Summary: For a partition \(\lambda\) of an integer, we associate \(\lambda\) with a slender poset \(P\) the Hasse diagram of which resembles the Ferrers diagram of \(\lambda\). Let \(X\) be the set of maximal chains of \(P\). We consider Stanley's involution \(\epsilon:X\rightarrow X\), which is extended from Schützenberger's evacuation on linear extensions of a finite poset [\textit{M.-P. Schützenberger}, Discrete Math. 2, 73--94 (1972; Zbl 0279.06001)]. We present an explicit characterization of the fixed points of the map \(\epsilon:X\rightarrow X\) when \(\lambda\) 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 a new Stembridge's \(q=-1\) phenomenon for the maximal chains of \(P\) under the involution \(\epsilon\) for the restricted shapes.