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

Jan Brandts, and Apo Cihangir. "Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group." Special Matrices 5.1 (2017): 158-201. <http://eudml.org/doc/288587>.

@article{JanBrandts2017,
abstract = {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 of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn [17] of symmetries of In. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’s theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler’s Tree of Fractions [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.},
author = {Jan Brandts, Apo Cihangir},
journal = {Special Matrices},
keywords = {Acute simplex; 0/1-matrix; Hadamard conjecture; hyperoctahedral group; cycle index; Pólya enumeration theorem; Kepler’s tree of fractions; strictly ultrametric matrix},
language = {eng},
number = {1},
pages = {158-201},
title = {Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group},
url = {http://eudml.org/doc/288587},
volume = {5},
year = {2017},
}

TY - JOUR
AU - Jan Brandts
AU - Apo Cihangir
TI - Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group
JO - Special Matrices
PY - 2017
VL - 5
IS - 1
SP - 158
EP - 201
AB - 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 of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn [17] of symmetries of In. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’s theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler’s Tree of Fractions [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.
LA - eng
KW - Acute simplex; 0/1-matrix; Hadamard conjecture; hyperoctahedral group; cycle index; Pólya enumeration theorem; Kepler’s tree of fractions; strictly ultrametric matrix
UR - http://eudml.org/doc/288587
ER -