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We introduce a new concept of permutation avoidance pattern called hattedpattern, which is a natural generalization of the barred pattern. We show thegrowth rate of the class of permutations avoiding a hatted pattern incomparison 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 hattedpattern avoiding permutations in $\s_n(132)$ by showing explicit bijections. Asa result, a new direct bijection between Motzkin paths and permutations in$\s_n(132)$ without two consecutive adjacent numbers is given. Thesepermutations are also represented on the Motzkin generating tree based on theEnumerative Combinatorial Object (ECO) method.

Combinatorial bijections from hatted avoiding permutations in $S_n(132)$ to generalized Dyck and Motzkin paths