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A055165
Number of invertible n X n matrices with entries equal to 0 or 1.
20
1, 1, 6, 174, 22560, 12514320, 28836612000, 270345669985440, 10160459763342013440
OFFSET
0,3
COMMENTS
All eigenvalues are nonzero.
LINKS
Eric Weisstein's World of Mathematics, Nonsingular Matrix.
Miodrag Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005; Linear Algebra and its Applications, 414 (2006), 310-346.
FORMULA
For an asymptotic estimate see A046747. A002884 is a lower bound. A002416 is an upper bound.
a(n) = n! * A088389(n). - Gerald McGarvey, Oct 20 2007
EXAMPLE
For n=2 the 6 matrices are {{{0, 1}, {1, 0}}, {{0, 1}, {1, 1}}, {{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}, {{1, 1}, {1, 0}}}.
PROG
(PARI) a(n)=sum(t=0, 2^n^2-1, !!matdet(matrix(n, n, i, j, (t>>(i*n+j-n-1))%2))) \\ Charles R Greathouse IV, Feb 09 2016
(Python)
from itertools import product
from sympy import Matrix
def A055165(n): return sum(1 for s in product([0, 1], repeat=n**2) if Matrix(n, n, s).det() != 0) # Chai Wah Wu, Sep 24 2021
CROSSREFS
Cf. A056990, A056989, A046747, A055165, A002416, A003024 (positive definite matrices).
A046747(n) + a(n) = 2^(n^2) = total number of n X n (0, 1) matrices = sequence A002416.
Main diagonal of A064230.
Sequence in context: A323279 A214801 A233225 * A318538 A071095 A134632
KEYWORD
nonn,nice,hard,more
AUTHOR
Ulrich Hermisson (uhermiss(AT)server1.rz.uni-leipzig.de), Jun 18 2000
EXTENSIONS
More terms from Miodrag Zivkovic (ezivkovm(AT)matf.bg.ac.rs), Feb 28 2006
Description improved by Jeffrey Shallit, Feb 17 2016
a(0)=1 prepended by Alois P. Heinz, Jun 18 2022
STATUS
approved