Motzkin paths and reduced decompositions for permutations with forbidden patterns
William Y. C. Chen, Yu P. Deng and Laura L. M. Yang
Abstract: We obtain a characterization of -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 -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 -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.
AMS Classification: 05A05, 05A15.
Keywords: Permutations with forbidden patterns, Motzkin paths, reduced decomposition, descents, strip decomposition, trapezoidal decomposition, inversion number.