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

Neil Sloane njasloane at gmail.com
Sat Mar 4 17:38:10 CET 2017

```Joerg, you are right. I should have stuck with
my ugly solutions, which were
1010
1010
0101
0101
and
0101
0101
1010
1010
Are there any other 4X4 0,1 matrices with A^2 = J = "all-ones" ?
Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com

On Sat, Mar 4, 2017 at 3:17 AM, Joerg Arndt <arndt at jjj.de> wrote:
> * 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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>> Seqfan Mailing list - http://list.seqfan.eu/
>
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```