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.