Restricted generating trees for weak orderings

TL;DR

This paper studies the enumeration of weak-ordering chains generated by restricted rooted trees, focusing on size 2 and 3 conditions, and explores their connections to pattern avoidance and descent statistics in permutations.

Contribution

It introduces a novel framework for counting weak-ordering chains via restricted trees and analyzes their combinatorial properties under specific size conditions.

Findings

Enumeration formulas for weak-ordering chains under size 2 and 3 conditions

Connections between these chains and pattern-avoiding permutations

Insights into descent statistics in related permutation classes

Abstract

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.