Motzkin paths and reduced decompositions for permutations with forbidden patterns (Q1408518)

Summary: We obtain a characterization of \((321, 3\bar{1}42)\)-avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of \((321,3\bar{1}42)\)-avoiding 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 \((231,4\bar{1}32)\)-avoiding 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.