Constrained Underdiagonal Paths and Pattern Avoiding Permutations

Moving from a simple bijection P between permutations Sn of length n and underdiagonal paths of size n, The we study and enumerate families of underdiagonal paths which are defined by restricting the bijection P to subclasses of Sn avoiding some vincular patterns. In particular, we will consider patterns of lengths 3 and 4, and, when it is possible, we will provide a characterization of the underdiagonal paths related to them in terms of geometrical constraints, or equivalently, the avoidance of some factors. Finally, we will provide a recursive growth of these families by means of generating trees and then their enumerative sequence.