Ordered Partitions and Drawings of Rooted Plane Trees

TL;DR

This paper explores the combinatorial structures of bounded regions in hyperplane arrangements, establishing bijections with ordered partitions and sequences of rooted plane trees, revealing new connections in geometric and combinatorial theory.

Contribution

It introduces novel bijections between bounded regions of hyperplane arrangements, ordered partitions with specific minima, and sequences of rooted plane trees, advancing understanding of these combinatorial objects.

Findings

Bijection between bounded regions and ordered partitions with odd left-to-right minima

Correspondence with sequences of rooted plane trees obtained by leaf removal

New insights into the structure of hyperplane arrangement regions

Abstract

We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with $n+1$ vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.