Repetition in reduced decompositions

AbstractGiven a permutation w, we show that the number of repeated letters in a reduced decomposition of w is always less than or equal to the number of 321- and 3412-patterns appearing in w. Moreover, we prove bijectively that the two quantities are equal if and only if w avoids the ten patterns 4321, 34 512, 45 123, 35 412, 43 512, 45 132, 45 213, 53 412, 45 312, and 45 231