Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns

We obtain a characterization of (321, 3¯ permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of (321, 3¯ permutations of length n equals the n-th Motzkin number, due to Gire, and further studied by Barcucci, Del Lungo, Pergola, Pinzani and Guibert. Similarly, we obtain a characterization of (231, 4 ¯ permutations. For these two classes, we show that the number of descents of a permutation equals the number of up steps on the corresponding Motzkin path. Moreover, we find a relationship between the inversion number of a permutation and the area of the corresponding Motzkin path.

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