Toric Richardson varieties of Catalan type and Wedderburn-Etherington numbers

TL;DR

This paper introduces a new class of toric Richardson varieties called of Catalan type, which are classified by binary trees and counted by Wedderburn-Etherington numbers, linking algebraic geometry with combinatorics.

Contribution

It defines Catalan type toric Richardson varieties, shows they are Fano Bott manifolds, and classifies them using binary trees and Wedderburn-Etherington numbers.

Findings

Toric Richardson varieties of Catalan type are classified by unordered binary trees.

The number of such varieties in dimension n equals the (n+1)th Wedderburn-Etherington number.

These varieties are Fano Bott manifolds.

Abstract

We associate a complete non-singular fan with a polygon triangulation. Such a fan appears from a certain toric Richardson variety, called of Catalan type introduced in this paper. A toric Richardson variety of Catalan type is a Fano Bott manifold. We show that toric Richardson varieties of Catalan type are classified up to isomorphism in terms of unordered binary trees. In particular, the number of isomorphism classes of $n$-dimensional toric Richardson varieties of Catalan type is the $(n+1)$th Wedderburn--Etherington number.