Explicit Formulae for Some Kazhdan-Lusztig R-Polynomials

We consider the Kazhdan-Lusztig polynomials Pu,v(q) indexed by permutations u, v having particular forms with regard to their monotonicity patterns. The main results are the following. First we obtain a simplified recurrence relation satisfied by Pu,v(q) when the maximum value of v ∈ Sn occurs in position n − 2 or n − 1. As a corollary we obtain the explicit expression for Pe,3 4 ... n 1 2(q) (where e denotes the identity permutation), as a q-analogue of the Fibonacci number. This establishes a conjecture due to M. Haiman. Second, we obtain an explicit expression for Pe,3 4 ... (n−2) n (n−1) 1 2(q). Our proofs rely on the recurrence relation satisfied by the Kazhdan-Lusztig polynomials when the indexing permutations are of the form under consideration, and on the fact that these classes of permutations lend themselves to the use of induction. We present several conjectures regarding the expression for Pu,v(q) under hypotheses similar to those of the main results.