Equivalence Classes of Full-Dimensional $$0/1$$ 0 / 1 -Polytopes with Many Vertices

Abstract

Let Q n denote the n-dimensional hypercube with the vertex set V n = {0, 1} n . A 0/1-polytope of Q n is a convex hull of a subset of V n . This paper is concerned with the enumeration of equivalence classes of full-dimensional 0/1-polytopes under the symmetries of the hypercube. With the aid of a computer program, Aichholzer completed the enumeration of equivalence classes of full-dimensional 0/1-polytopes for Q 4 , Q 5 , and those of Q 6 up to 12 vertices. In this paper, we present a method to compute the number of equivalence classes of full-dimensional 0/1-polytopes of Q n with more than 2 n−3 vertices. As an application, we finish the counting of equivalence classes of full-dimensional 0/1-polytopes of Q 6 with more than 12 vertices.

Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications

“…Hypercubes [1]- [29] and related combinatorics of wreath product groups [30]- [54] have been the focus of a number of research investigations owing to their importance in numerous applications in a variety of disciplines. Hypercubes are natural representations of Boolean functions, as 2n possible Boolean functions from a set of n entities that take binary values can be represented by the vertices of a hypercube.…”

“…Historically Pólya's theorem was anticipated in Redfield's paper on superposition theorem [16]. Although in more recent mathematical literature, cycle indices of hypercubes and enumerations of colorings of the vertices of hypercubes have been considered [17]- [29], [34] these studies have been restricted only to the totally symmetric irreducible representations of the hyperoctahedral groups. Moreover in the most recent work on the 5D-hypercube enumeration [29] of vertex colorings there are errors, as we show here.…”

“…Although in more recent mathematical literature, cycle indices of hypercubes and enumerations of colorings of the vertices of hypercubes have been considered [17]- [29], [34] these studies have been restricted only to the totally symmetric irreducible representations of the hyperoctahedral groups. Moreover in the most recent work on the 5D-hypercube enumeration [29] of vertex colorings there are errors, as we show here. Pólya's theorem and its variation [1]- [6], [17]- [21] have been applied extensively which generate equivalence classes for different distribution of colors called the pattern inventory and also the total number of colorings.…”

“…Each such hyperplane generates a set of cycle types for each conjugacy class. Thus computing the equivalence classes of the colorings of various hyperplanes requires the computation of the cycle types of different (n-q)-hyperplanes of the hypercube with q=1 through n. Previous works in the mathematical literature [17]- [29] have focused on the total number of equivalence classes rather than the inventory of patterns or a generating function that yields number of colorings for a given number of colors of various kinds. Such a distribution of patterns for various colors is quite important for a number of practical applications, and thus we focus in the present study the computational techniques to obtain such generating functions for all hyperplanes and all irreducible representations of the hypercube.…”

“…Such a distribution of patterns for various colors is quite important for a number of practical applications, and thus we focus in the present study the computational techniques to obtain such generating functions for all hyperplanes and all irreducible representations of the hypercube. Moreover none of the previous studies [17]- [29] has dealt with irreducible representations other than totally symmetric representation in their enumerations. The present author [11] has previously considered multinomial colorings of 4D-hypercube for different hyperplanes, and with chemical applications to water pentamer in mind, the present author has considered colorings of tesseracts [64] of the 5D-hypercube, and recently vertices (q=4) and tesseracts q=1 for all irreducible representations and 2-colorings of (q=2) 3-faces only for the totally symmetric irreducible representation of the 5D-hypercube [61].…”