Kazhdan–Lusztig R-polynomials of permutations containing a 231 or 312 pattern

We consider the Kazhdan–Lusztig R  -polynomials, Ru,v(q)Ru,v(q), indexed by permutations u,vu,v of SnSn, where u   contains a pattern of type 231 or 312 and vv is obtained from u by applying a transposition and an appropriate 3-cycle. We prove, using combinatorial techniques, that these polynomials are given by a closed product formula. The case of permutations which contain a pattern of type 321 follows from the others.