The Bruhat Order on Symmetric Groups via Intrinsic Coverings of Compositions

TL;DR

This paper characterizes the strong Bruhat order on symmetric groups using an intrinsic poset structure on staircase compositions derived from Lehmer's code, avoiding direct permutation references.

Contribution

It introduces a new intrinsic combinatorial characterization of the Bruhat order via compositions, independent of permutation representations.

Findings

Poset structure on compositions is equivalent to the Bruhat order

Provides an intrinsic, permutation-free description of the Bruhat order

Enhances combinatorial understanding of symmetric group orderings

Abstract

Lehmer's code defines a bijection between the symmetric group and the set of staircase compositions. In this paper, we characterize a poset structure on these compositions that is equivalent to the strong Bruhat order on the symmetric group. This construction is intrinsic and does not require any reference to the associated permutations.