Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations

In (Deodhar, Geom. Dedicata, 36(1) (1990), 95-119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials P in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where W=G(n) (the symmetric group on n letters) and the permutation w is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for w. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincare polynomial of the intersection cohomology of the Schubert variety corresponding to w is (1+q)(l(w)) if and only if w is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety X to have a small resolution. We conclude with a simple method for completely determining the singular locus of X when w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (B-n, F-4, G(2)).