Simion's type $B$ associahedron is a pulling triangulation of the Legendre polytope

TL;DR

This paper demonstrates that Simion's type B associahedron can be realized as a pulling triangulation of the Legendre polytope, revealing new combinatorial and geometric structures and symmetries.

Contribution

It establishes a combinatorial equivalence between Simion's type B associahedron and a pulling triangulation of the Legendre polytope, extending symmetry actions and connecting to Delannoy paths.

Findings

Simion's type B associahedron is a pulling triangulation of the Legendre polytope.

Every pulling triangulation of the Legendre polytope yields a flag complex.

A bijection between faces of the associahedron and Delannoy paths is provided.

Abstract

We show that Simion's type $B$ associahedron is combinatorially equivalent to a pulling triangulation of a type $B$ root polytope called the Legendre polytope. Furthermore, we show that every pulling triangulation of the Legendre polytope yields a flag complex. Our triangulation refines a decomposition of the Legendre polytope given by Cho. We extend Cho's cyclic group action to the triangulation in such a way that it corresponds to rotating centrally symmetric triangulations of a regular $(2n+2)$-gon. Finally, we present a bijection between the faces of the Simion's type $B$ associahedron and Delannoy paths.