Rectangulotopes

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 fin, 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. (c) 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.