Rectangulotopes

TL;DR

This paper introduces polytopes called quotientopes related to rectangulations, providing explicit descriptions and coordinate formulas, connecting combinatorial structures with geometric realizations.

Contribution

It presents new realizations of polytopes associated with rectangulations, including explicit vertex and facet descriptions and coordinate formulas, linking combinatorics and geometry.

Findings

Defined quotientopes for rectangulations

Provided explicit vertex and facet descriptions

Derived formulas for vertex coordinates

Abstract

Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles. They are defined as quotientopes of natural lattice congruences on the weak Bruhat order on permutations in $\mathfrak{S}_n$, and their skeleta are flip graphs on rectangulations. We give simple vertex and facet descriptions of these polytopes, in particular elementary formulas for computing the coordinates of the vertex corresponding to each rectangulation, in the spirit of J.-L. Loday's realization of the associahedron.