Fixed Points of the Evacuation of Maximal Chains on Fuss Shapes

Sen-Peng Eu, Tung-Shan Fu, Hsiang-Chun Hsu, Yu-Pei Huang


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 Sch├╝tzenberger's evacuation on linear extensions of a finite poset. 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 anew Stembridge's $q=-1$ phenomenon for the maximal chains of $P$ under the involution $\epsilon$ for the restricted shapes.


Evacuation; Stembridge's q=-1 phenomenon; Fixed point; Partition

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