  We give a combinatorial formula for the Kazhdan-Lusztig polynomials $P_{x,w}$
in the symmetric group when $w$ is a 321-hexagon-avoiding permutation. Our
formula, which depends on a combinatorial framework developed by Deodhar, can
be expressed in terms of a simple statistic on all subexpressions of any fixed
reduced expression for $w$. We also show that $w$ being 321-hexagon-avoiding is
equivalent to several other conditions, such as the Bott-Samelson resolution of
the Schubert variety $X_w$ being small. We conclude with a simple method for
completely determining the singular locus of $X_w$ when $w$ is
321-hexagon-avoiding.
