Restricted generating trees for weak orderings

Motivated by the study of pattern avoidance in the context of permutations and ordered partitions, we consider the enumeration of weak-ordering chains obtained as leaves of certain restricted rooted trees. A tree of order n is generated by inserting a new variable into each node at every step. A node becomes a leaf either after n steps or when a certain stopping condition is met. In this paper we focus on conditions of size 2 (x = y, x < y, or x ≤ y) and several conditions of size 3. Some of the cases considered here lead to the study of descent statistics of certain 'almost' pattern-avoiding permutations.