# On the Reduciblity of a Certain Type of Rank 3 Uniform Oriented Matroid by a Point

## TL;DR

This paper investigates the combinatorial structure of regions formed by sides and diagonals of convex polygons, linking them to cycles in the symmetric group, and characterizes when certain oriented matroids are reducible based on these regions.

## Contribution

It introduces a novel association between polygon regions and specific cycles in the symmetric group, providing a combinatorial characterization of reducible rank 3 uniform oriented matroids.

## Key findings

- Most cycles do not occur in any given convex n-gon.
- Characterization of cycles that always occur in every convex n-gon.
- Identification of conditions under which a one point extension of a uniform rank 3 oriented matroid is reducible.

## Abstract

For a positive integer $n\geq 3$, the sides and diagonals of a convex $n$-gon divide the interior of the convex $n$-gon into finitely (polynomial in $n$) many regions bounded by them. In this article, we associate to every region a unique $n$-cycle in the symmetric group $S_n$ of a certain type (defined as $2$-standard consecutive cycle) by studying point arrangements in the plane. Then we find that there are more (exponential in $n$) number of such cycles leading to the conclusion that not every region labelled by a cycle appears in every convex $n$-gon. In fact most of them do not occur in any given single convex $n$-gon. Later in the main theorem of this article we characterize combinatorially those cycles (defined as definite cycles) whose corresponding regions occur in every convex $n$-gon and those cycles (defined as indefinite cycles) whose corresponding regions do not occur in every convex $n$-gon. As a consequence we characterize those one point extensions of a uniform rank $3$ convex oriented matroid for which the one point extension is reducible by the, one point, when it lies inside the convex hull.