Enumerative combinatorics
Kazhdan-Lusztig polynomials for certain varieties of incomplete flags
Proceedings of the 7th conference on Formal power series and algebraic combinatorics
Kazhdan-Lusztig and R-polynomials from a combinatorial point of view
Discrete Mathematics - selected papers in honor of Adriano Garsia
Proof of Two Conjectures of Brenti and Simion on Kazhdan-Lusztig Polynomials
Journal of Algebraic Combinatorics: An International Journal
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We consider the Kazhdan-Lusztig polynomials P_u, 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 P_u, v(q) when the maximum value of v ∈ S_n occurs in position n − 2 or n − 1. As a corollary we obtain the explicit expression for P_e, 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 P_e, 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 P_u, v(q) under hypotheses similar to those of the main results.