# Proofs of Conjectures about Pattern-Avoiding Linear Extensions

## TL;DR

This paper investigates pattern avoidance in linear extensions of finite posets, proving conjectures related to k-ary heaps and rectangular posets, advancing understanding of permutation patterns in poset theory.

## Contribution

It proves a conjecture about pattern avoidance in k-ary heaps and confirms several conjectures on linear extensions of rectangular posets.

## Key findings

- Proved a conjecture on pattern avoidance in k-ary heaps.
- Confirmed conjectures about pattern-avoiding linear extensions of rectangular posets.
- Established general results linking poset linear extensions to permutation pattern avoidance.

## Abstract

After fixing a canonical ordering (or labeling) of the elements of a finite poset, one can associate each linear extension of the poset with a permutation. Some recent papers consider specific families of posets and ask how many linear extensions give rise to permutations that avoid certain patterns. We build off of two of these papers. We first consider pattern avoidance in $k$-ary heaps, where we obtain a general result that proves a conjecture of Levin, Pudwell, Riehl, and Sandberg in a special case. We then prove some conjectures that Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan made about pattern-avoiding linear extensions of rectangular posets.