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 u d … d u and Motzkin paths via weakly avoiding permutations in S n ( 132 ) .