  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.

Combinatorics