

A283627


The number of (n^2) X (n^2) real {0,1}matrices the square of which is the allones matrix.


2




OFFSET

1,2


COMMENTS

These are real {0,1} matrices A such that A^2 = J, the allones matrix.
It is known that A must have dimension which is a square, so the sequence shows counts for 1 X 1, 4 X 4, 9 X 9, 16 X 16 matrices and so on.
It is also known that if A has dimension n^2 X n^2, then each row and column must contain exactly n 1's, and A has trace n.
a(3) = 1330560 is confirmed by W. Edwin Clark, Mar 12 2017, who says: (Start)
My method was to take the 6 matrices A1, A2, A3, A4, A5, A6 found by Knuth, which are representatives for the 6 distinct orbits of 9 x 9 matrices A such that A^2 = J under the action of the 9! permutation matrices acting by conjugation.
I found for each Ai the size n_i of the stabilizer of Ai. The stabilizer orders are [n_1,n_2,n_3,n_4,n_5,n_6] = [6,2,1,1,2,2], which implies that the cardinality of the union of all orbits is Sum(9!/n_i, i=1..6) = 1330560.
(End)


LINKS



EXAMPLE

Four of the 12 solutions for 4 X 4 are
1 1 0 0
0 0 1 1
1 1 0 0
0 0 1 1
.
1 1 0 0
0 0 1 1
0 0 1 1
1 1 0 0
.
1 0 0 1
0 1 1 0
1 0 0 1
0 1 1 0
.
0 1 0 1
1 0 1 0
1 0 1 0
0 1 0 1
.
Solutions for 9 X 9 are, for example,
1 1 1 0 0 0 0 0 0
0 0 0 1 0 1 1 0 0
0 0 0 0 1 0 0 1 1
1 1 1 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0
0 0 0 1 0 1 1 0 0
0 0 0 0 1 0 0 1 1
0 0 0 1 0 1 1 0 0
0 0 0 0 1 0 0 1 1
.
1 1 1 0 0 0 0 0 0
0 0 0 1 0 1 1 0 0
0 0 0 0 1 0 0 1 1
1 1 1 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0
0 0 0 1 0 1 1 0 0
0 0 0 0 1 0 0 1 1
0 0 0 1 0 0 0 1 1
0 0 0 0 1 1 1 0 0
.
1 1 1 0 0 0 0 0 0
0 0 0 1 0 1 1 0 0
0 0 0 0 1 0 0 1 1
1 1 1 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0
0 0 0 1 0 1 0 0 1
0 0 0 0 1 0 1 1 0
0 0 0 1 0 1 0 0 1
0 0 0 0 1 0 1 1 0


CROSSREFS



KEYWORD

nonn,bref,more


AUTHOR



EXTENSIONS



STATUS

approved



