An order on circular permutations

TL;DR

This paper introduces a ranked poset structure on circular permutations derived from affine Weyl groups, revealing its lattice properties, rank function, and connections to combinatorial objects like Eulerian numbers and Young's lattice.

Contribution

It defines a new poset on circular permutations linked to affine Weyl groups, proving its semidistributive lattice structure and computing its rank function.

Findings

The poset is a semidistributive lattice.

The rank function is explicitly computed using inversions.

Connections to Eulerian numbers, polygon triangulations, and Young's lattice are established.

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.