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The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory and the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no completely combinatorial interpretation for them is known in general. Deodhar has given a framework which generally involves recursion to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a simple combinatorial formula for certain Kazhdan-Lusztig polynomials of finite Weyl groups. This generalizes results of Billey-Warrington which identified the 321-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. This is joint work with Sara Billey.