Q. about {0,1}-matrices.

[N. J. A. Sloane]

I didn't do anything like that since I decided in advanced that I
wouldn't spend more than 15 mins programming (given that I had a
determinant function already).

However, you are right. I guess that those 2224232640 robust
matrices of order 6 lie in slightly fewer than 200,000 equivalence
classes. The actual value is the number of equivalence classes of
nonsingular 0-1 matrices of order 6 - I can't find that sequence
in OEIS, is it there?

That would reduce the time for a(7) to a few hours at most.
The number of equivalence classes for order 7 would be about
25 million or so; still (just) within the feasible range.

Incidentally, the definition of A006383 (Number of equivalence
classes of n X n binary matrices when S_n acts independently on the
rows and columns) seems to be the same as the definition of A002724
(Number of inequivalent n X n binary matrices, where equivalence
means permutations of rows and/or columns) but the numbers are
different - what gives?  A002724 is the one that seems to have
values fitting the definition.

Brendan.


* N. J. A. Sloane <njas at research.att.com> [070107 10:50]:
> Brendan said:
> 
> In a little less than 30 minutes, my C program counted:
> 
>    1, 4, 68, 5008, 1603232, 2224232640
> 
> Me:  Pretty amazing!   
> 
> Did your C program use the following observation:
>     - the number of ways to add a border to an n-1 X n-1
>       matrix A doesn'y change if you permute the rows
>       and columns of A
> 
> For going from n=6 to n=7 that's a potential saving
> of 6! * 6!.  In other words, for each of the 2224232640
> 6x6s, put it into canonical form.
> When you have all the canon. forms, and their multiplicities,
> then see how many ways each can be extended.
> But you are the Latin square expert, so probably you did this
> without thinking it was worth mentioning.
> 
> Neil