Geodesic
Reduced decompositions and permutation patterns
Journal of Algebraic Combinatorics, Tome 24 (2006) no. 3, pp. 263-284.

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Summary: Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a property of their reduced decompositions. This paper generalizes that result with a detailed study of permutations via their reduced decompositions and the notion of pattern containment. These techniques are used to prove a new characterization of vexillary permutations in terms of their principal dual order ideals in a particular poset. Additionally, the combined frameworks yield several new results about the commutation classes of a permutation. In particular, these describe structural aspects of the corresponding graph of the classes and the zonotopal tilings of a polygon defined by Elnitsky that is associated with the permutation.
Keywords: keywords reduced decomposition, permutation pattern, vexillary permutation, zonotopal tiling, freely braided permutation 
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Tenner, Bridget Eileen. Reduced decompositions and permutation patterns. Journal of Algebraic Combinatorics, Tome 24 (2006) no. 3, pp. 263-284. http://geodesic.mathdoc.fr/item/JAC_2006__24_3_a4/