The range of repetition in reduced decompositions

Given a permutation w, we look at the range of how often a simple reflection s_k appears in reduced decompositions of w. We compute the minimum and give a sharp upper bound on the maximum. That bound is in terms of 321- and 3412-patterns in w, specifically as they relate in value and position to k. We also characterize when that minimum and maximum are equal, refining a previous result that braid moves are equivalent to 321-patterns.

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