OFFSET
1,1
COMMENTS
This majorizes the sequence of maximal determinants only up to the 6th term. It is conjectured that the sequence of maximal determinants majorizes this for all later terms.
The first term needing verification is a(11) >= 739. a(12) = 2173 has been verified by Brent, Orrick, Osborn, and Zimmermann in 2010. Lower bounds for the next terms: a(13) >= 6739, a(14) >= 21278, a(15) >= 69259, a(16) >= 230309. - Hugo Pfoertner, Jan 03 2020
LINKS
Swee Hong Chan and Igor Pak, Computational complexity of counting coincidences, arXiv:2308.10214 [math.CO], 2023. See p. 18.
R. Craigen, The Range of the Determinant Function on the Set of n X n (0,1)-Matrices, J. Combin. Math. Combin. Computing, 8 (1990) pp. 161-171.
William P. Orrick, The maximal {-1, 1}-determinant of order 15, arXiv:math/0401179 [math.CO], 2004.
William P. Orrick, Spectrum of the determinant function.
G. R. Paseman, Related Material
Miodrag Zivkovic, Massive computation as a problem solving tool, in Proceedings of the 10th Congress of Yugoslav Mathematicians (Belgrade, 2001), pages 113-128. Univ. Belgrade Fac. Math., Belgrade, 2001.
M. Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
EXAMPLE
There is no 3 X 3 {0,1}-matrix with determinant 3, as such a matrix must have a row with at least one 0 in it.
PROG
(Python)
from itertools import product
from sympy import Matrix
def A013588(n):
s, k = set(Matrix(n, n, p).det() for p in product([0, 1], repeat=n**2)), 1
while k in s:
k += 1
return k # Chai Wah Wu, Oct 01 2021
CROSSREFS
KEYWORD
nice,more,hard,nonn
AUTHOR
Gerhard R. Paseman (paseman(AT)prado.com)
EXTENSIONS
Extended by William Orrick, Jan 12 2006. a(7), a(8) and a(9) computed by Miodrag Zivkovic. a(7) and a(8) independently confirmed by Antonis Charalambides. a(10) computed by William Orrick.
STATUS
approved