Formulas for multi-parameter classes of Kazhdan–Lusztig polynomials in S(n)S(n)

We give explicit formulas for several classes of Kazhdan–Lusztig polynomials of the symmetric group which are related to others already considered in the literature. In particular, we generalize two theorems of Brenti and Simion [Explicit formulae for some Kazhdan–Lusztig polynomials, J. Algebraic Combin. 11 (2000) 187–196], we prove an unpublished conjecture of Brenti, and we treat the case missing in Theorem 4.8 of F. Caselli [Proof of two conjectures of Brenti and Simion on Kazhdan–Lusztig polynomials, J. Algebraic Combin. 18 (2003) 171–187].