Ordered Partitions and Drawings of Rooted Plane Trees
Qingchun Ren

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 consisting of the hyperplanes defined by and . The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with 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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Graph Theory Research · Computational Geometry and Mesh Generation
