[seqfan] Re: Number of nXn (0,1)-matrices A with A^2 = J?
W. Edwin Clark
wclark at mail.usf.edu
Sat Mar 4 20:31:42 CET 2017
I get the following 12 4x4 0,1 matrices A with A^2 = J.
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
0 0 1 1
1 1 0 0
1 1 0 0
0 0 1 1
0 0 1 1
1 1 0 0
0 0 1 1
1 1 0 0
1 0 1 0
1 0 1 0
0 1 0 1
0 1 0 1
1 0 1 0
0 1 0 1
0 1 0 1
1 0 1 0
0 1 0 1
1 0 1 0
1 0 1 0
0 1 0 1
0 1 0 1
0 1 0 1
1 0 1 0
1 0 1 0
0 1 1 0
0 1 1 0
1 0 0 1
1 0 0 1
0 1 1 0
1 0 0 1
0 1 1 0
1 0 0 1
1 0 0 1
0 1 1 0
1 0 0 1
0 1 1 0
1 0 0 1
1 0 0 1
0 1 1 0
0 1 1 0
On Sat, Mar 4, 2017 at 11:38 AM, Neil Sloane <njasloane at gmail.com> wrote:
> 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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> >
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