Proof of two conjectures of Brenti and Simion on Kazhdan-Lusztig polynomials

We find an explicit formula for the Kazhdan-Lusztig polynomials Pu-i,Pu-a,v(i) of the symmetric group G(n) where, for a, i, n is an element of N such that 1less than or equal toaless than or equal toiless than or equal ton, we denote by u(i,a)=s(a)s(a+1)...s(i-1) and by v(i) the element of G(n) obtained by inserting n in position i in any permutation of G(n-1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two conjectures of Brenti and Simion [4, Conjectures 4.2 and 4.3], and includes as a special case a result of Shapiro, Shapiro and Vainshtein [13, Theorem 1]. All the proofs are purely combinatorial and make no use of the geometry of the corresponding Schubert varieties.