Geometric view of interval poset permutations

TL;DR

This paper introduces a geometric perspective on interval posets of permutations, establishing a bijection with polygon dissections, and explores their structural properties and connections.

Contribution

It provides a novel geometric framework linking interval posets of permutations to polygon dissections, enriching the combinatorial understanding.

Findings

Established a bijection between interval posets and polygon dissections.

Identified special subsets of interval posets with particular polygon dissection properties.

Uncovered new combinatorial connections between permutations and geometric structures.

Abstract

In a recent study by Tenner, the concept of the interval poset of a permutation was introduced to effectively represent all intervals and their inclusions within a permutation. In this paper, we present a new geometric viewpoint on interval posets. We establish a one-to-one correspondence between the set of interval posets for permutations of size n and a specific subset of dissections of a convex polygon with n sides. Through this correspondence, we investigate various intriguing subsets of interval posets and uncover their connections with specific polygon dissections.