  Not long ago, Claesson and Mansour proposed some conjectures about the
enumeration of the permutations avoiding more than three Babson - Steingr\'\i
msson patterns (generalized patterns of type $(1,2)$ or $(2,1)$). The avoidance
of one, two or three patterns has already been considered. Here, the cases of
four and five forbidden patterns are solved and the exact enumeration of the
permutations avoiding them is given, confirming the conjectures of Claesson and
Mansour. The approach we use can be easily extended to the cases of more than
five forbidden patterns.
