Permutations weakly avoiding barred patterns and combinatorial bijections to generalized Dyck and Motzkin paths

We explore the weak avoidance of barred patterns in permutations introduced recently by J.-L. Baril. We give its general properties in comparison with the barred pattern avoidance. By showing explicit bijections, we revisit Dyck paths with no peak at height p, Dyck paths with no ud…du and Motzkin paths via weakly avoiding permutations in Sn(132).