Fixed points of the evacuation of maximal chains on fuss shapes

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

The electronic journal of combinatorics, Tome 25 (2018) no. 1

Résumé

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. 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.

Détail

DOI : 10.37236/6898

Classification : 05A15, 05A19
Mots-clés : evacuation, promotion, slender posets, linear extensions, maximal chains, cyclic sieving phenomenon