An equivalence relation on the symmetric group and multiplicity-free flag $h$-vectors

Abstract

We consider the equivalence relation ∼ on the symmetric group Sn generated by the interchange of any two adjacent elements ai and ai+1 of w = a1 · · · an ∈ Sn such that |ai − ai+1| = 1. We count the number of equivalence classes and the sizes of the equivalence classes. The results are generalized to permutations of multisets. In the original problem, the equivalence class containing the identity permutation is the set of linear extensions of a certain poset. Further investigation yields a characterization of all finite graded posets whose flag h-vector takes on only the values 0,±1.