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

TL;DR

This paper introduces an equivalence relation on the symmetric group based on adjacent transpositions with difference one, counts classes, and characterizes posets with specific flag h-vector values, extending to multisets.

Contribution

It defines a new equivalence relation on permutations, counts classes, and characterizes certain finite graded posets with restricted flag h-vectors, generalizing results to multisets.

Findings

Counted the number of equivalence classes in S_n.

Characterized all finite graded posets with flag h-vectors in {-1,0,1}.

Extended results to permutations of multisets using umbral techniques.

Abstract

We consider the equivalence relation ~ on the symmetric group S_n generated by the interchange of two adjacent elements a_i and a_{i+1} of w=a_1 ... a_n in S_n such that |a_i - a_{i+1}|=1. We count the number of equivalence classes and the sizes of equivalence classes. The results are generalized to permutations of multisets using umbral techniques. 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 -1, 0, 1.