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Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

Jan Brandts;  Apo Cihangir — 2017

Special Matrices

The convex hull of n + 1 affinely independent vertices of the unit n-cube In is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. The first part of this paper deals with giving a detailed description...

Counting triangles that share their vertices with the unit n -cube

Brandts, Jan;  Cihangir, Apo — 2013

Applications of Mathematics 2013

This paper is about 0 / 1 -triangles, which are the simplest nontrivial examples of 0 / 1 -polytopes: convex hulls of a subset of vertices of the unit n -cube I n . We consider the subclasses of right 0 / 1 -triangles, and acute 0 / 1 -triangles, which only have acute angles. They can be explicitly counted and enumerated, also modulo the symmetries of I n .

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