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arXiv · 2010.06528

An order on circular permutations

Antoine Abram·Nathan Chapelier-Laget·Christophe Reutenauer
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Abstract

Motivation coming from the study of affine Weyl groups, a structure of ranked poset is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

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Antoine Abram, Nathan Chapelier-Laget, Christophe Reutenauer. 2020-10-13. An order on circular permutations. https://arxiv.org/abs/2010.06528

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