Enumeration of bigrassmannian permutations below a permutation in Bruhat order

TL;DR

This paper derives formulas for counting bigrassmannian permutations below a given permutation in Bruhat order within symmetric groups, utilizing characterizations by Lascoux-Schutzenberger and Reading.

Contribution

It provides new enumeration formulas for bigrassmannian permutations in Bruhat order, connecting combinatorial characterizations with algebraic enumeration.

Findings

Formulas for counting bigrassmannian permutations below a permutation in Bruhat order

Use of characterizations by Lascoux-Schutzenberger and Reading

Enhanced understanding of the structure of Coxeter groups

Abstract

In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.