Geodesic
Motzkin paths and reduced decompositions for permutations with forbidden patterns
The electronic journal of combinatorics, Permutation Patterns, Tome 9 (2002) no. 2
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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.
DOI : 10.37236/1687
Classification : 05A05, 05A15 
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     author = {William Y. C. Chen and Yu-Ping Deng and Laura L. M. Yang},
     title = {Motzkin paths and reduced decompositions for permutations with forbidden patterns},
     journal = {The electronic journal of combinatorics},
     year = {2002},
     volume = {9},
     number = {2},
     doi = {10.37236/1687},
     zbl = {1023.05002},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1687/}
}
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William Y. C. Chen; Yu-Ping Deng; Laura L. M. Yang. Motzkin paths and reduced decompositions for permutations with forbidden patterns. The electronic journal of combinatorics, Permutation Patterns, Tome 9 (2002) no. 2. doi: 10.37236/1687

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