Proof of Two Conjectures of Brenti and Simion on Kazhdan-Lusztig Polynomials

We find an explicit formula for the Kazhdan-Lusztig polynomials Pui a, vi of the symmetric group fraktur G sign(n) where, for a, i, n ∈ ℕ such that 1 ≤ a ≤ i ≤ n, we denote by ui,a = s asa+1 script G sign si-1 and by vi the element of fraktur G sign(n) obtained by inserting n in position i in any permutation of fraktur G sign(n - 1) allowed to lise 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.