Interval Posets and Polygon Dissections

TL;DR

This paper explores the geometric interpretation of interval posets of permutations, establishing a bijection with convex polygons with non-crossing diagonals, and applies this to enumeration and permutation analysis.

Contribution

It introduces a novel geometric perspective on interval posets via a bijection with convex polygons, enhancing understanding and enumeration methods.

Findings

Bijection between tree interval posets and convex polygons with non-crossing diagonals

Enumeration formulas for interval posets using geometric interpretation

Application to block-wise simple permutations

Abstract

The Interval poset of a permutation is an effective way of capturing all the intervals of the permutation and the inclusions between them and was introduced recently by Tenner. Thi paper explores the geometric interpretation of interval posets of permutations. We present a bijection between tree interval posets and convex polygons with non-crossing diagonals, offering a novel geometric perspective on this purely combinatorial concept. Additionally, we provide an enumeration of interval posets using this bijection and demonstrate its application to block-wise simple permutations.