An order on circular permutations (Q2048566)

Summary: Motivated by the study of affine Weyl groups, a ranked poset structure 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.