### Abstract

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.

Original language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 25 |

Issue number | 1 |

Publication status | Published - 2018 Feb 16 |

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### Keywords

- Cyclic sieving phenomenon
- Evacuation
- Linear extensions
- Maximal chains
- Promotion
- Slender posets

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*25*(1).