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 $x\le y$) 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.

• 05A05 - Permutations, words, matrices
• 05A15 - Exact enumeration problems, generating functions
• 05A18 - Partitions of sets
• 05C05 - Trees
• 06A07 - Combinatorics of partially ordered sets

• 1 zbMATH Open