Poincar\'e polynomial for fully commutative elements in the symmetric group

TL;DR

This paper computes the Poincaré polynomial for fully commutative elements in the symmetric group, providing a closed-form generating function that captures their combinatorial structure.

Contribution

It derives an explicit formula for the generating function of fully commutative elements in the symmetric group, advancing understanding of their algebraic and combinatorial properties.

Findings

Derived a closed-form expression for a_n(q)

Established connections between fully commutative elements and Coxeter group structure

Enhanced enumeration techniques for Coxeter group elements

Abstract

Let $W^c(A_n)$ be the set of fully commutative elements of the Coxeter group $W(A_n)$. Let $$ a_n(q)= \sum_{w \in W^c(A_n)} q^{l(w)} . $$ We compute $a_n(q)$.