# [seqfan] Re: Number of nXn (0,1)-matrices A with A^2 = J?

Joerg Arndt arndt at jjj.de
Sat Mar 4 09:17:13 CET 2017

```* Neil Sloane <njasloane at gmail.com> [Mar 04. 2017 09:00]:
> I just came across an old paper of Herb Ryser (A generalization
> of the matrix equation A^2=J, Linear Alg. Applic., 3 (1970),451-460)
> where he mentions in passing the unsolved problem of finding
> the number of n X n real (0,1) matrices A such that A^2 is the
> all-ones matrix J.
>
> He says that A must have constant row and column sums c, and c^2 = n.
> Also trace A = c. So n must be a square, n=c^2.
>
> There is a long entry in the Index to the OEIS under
> matrices, binary
> but this problem doesn't seem to be mentioned there.
> Is this sequence in the OEIS?  (If so, it should
> be mentioned in the index entry.)
> I'm pretty sure I've seen the problem before, but a quick
> search in the OEIS didn't find it.  Let n = c^2. Then there is one
> solution if c=1, and if someone could work out the answers for c=2 and
> maybe 3, that might be enough to locate it.
>
> Here is one solution for c=2:
> 1010
> 0101
> 1010
> 0101

This one has trace = c^2 = 4, not c as stated above.

> and permuting the rows is allowed - so is the answer 6 if c=2?
>
> Neil
>
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```