Explicit Formulae for Some Kazhdan–Lusztig R-Polynomials

We consider the Kazhdan–Lusztig R-polynomials, Ru, v(q) indexed by permutations “u, v” having particular forms. More precisely, we show that Re, 34…n12(q) (where “e” denotes the identity permutation) equals, aside from a simple change of variable, a q-analogue of the Fibonacci number, and if two permutations are obtained one from the other by applying two transpositions (one simple, and one not), then the corresponding R-polynomial factors nicely. Our proofs are combinatorial.