Dyck paths and restricted permutations

This paper is devoted to characterize permutations with forbidden patterns by using canonical reduced decompositions, which leads to bijections between Dyck paths and Sn(321)Sn(321) and Sn(231)Sn(231), respectively. We also discuss permutations in SnSn avoiding two patterns, one of length 3 and the other of length k. These permutations produce a kind of discrete continuity between the Motzkin and the Catalan numbers.