Combinatorial bijections from hatted avoiding permutations in $S_n(132)$   to generalized Dyck and Motzkin paths
  We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to barred pattern. We prove that Dyck paths with no peak at height $p$, Dyck paths with no $ud... du$ and Motzkin paths are counted by hatted pattern avoiding permutations in $\s_n(132)$ by showing explicit bijections. As a result, a new direct bijection between Motzkin paths and permutations in $\s_n(132)$ without two consecutive adjacent numbers is given. These permutations are also represented on the Motzkin generating tree based on the Enumerative Combinatorial Object (ECO) method. 
Phan Thuan Do, Dominique Rossin and Thi Thu Huong Tran
math.CO