Explicit formulae for some Kazhdan-Lusztig R-polynomials

We consider the Kazhdan Lusztig R-polynomials, R-u,R- e (q) indexed by permutations "u, v" having particular forms. More precisely we show that R-e,R-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 permututions are obtained one from the other by applying two transpositions tone simple. and one not I, then the corresponding R-polynomial factors nicely. Our proof are combinatorial. (C) 2001 Academic Press.