Regions of the type C Catalan arrangement

TL;DR

This paper establishes a bijection between certain labeled forests and the regions of the type C Catalan arrangement, providing a combinatorial proof of their enumeration formula.

Contribution

It introduces a new bijective proof linking rooted labeled forests with regions of the type C Catalan arrangement, enhancing combinatorial understanding.

Findings

Bijection between labeled forests and arrangement regions

Proof of the enumeration formula for regions

Enhanced combinatorial interpretation of type C Catalan regions

Abstract

In this paper, we give a bijection between rooted labeled ordered forests with a selected subset of their leaves and the regions of the type $C$ Catalan arrangement in $\R^n$. We thus obtain a bijective proof of the well-known enumeration formula of these regions ${2^n}n! \binom{2n}{n}$.