The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group $S_n$

In this paper we show that the leading coefficient $\mu (y,w)$ of certain Kazhdan-Lusztig polynomials $P_{y,w}$ of the permutation group ${\mathfrak{S}}_n$ of 1,2,…,n are not greater than 1. More precisely, we show that the leading coefficients $\mu (y,w)$ are not greater than 1 whenever $a(y)<a(w)$, where $a: \mathfrak S_n\to\mathbf N$ is the function defined by Lusztig.