A085000 and related seqs

[all at abouthugo.de]

SeqFans,

%S A085000 1,10,412,40800,6839492,1865999570
%N A085000 Maximal determinant of an n X n matrix using the integers 1
to n^2.

is a sequence I've found extremely interesting after stumbling upon it
more or less by chance when browsing through "bref" sequences.
At that time it only had the first 3 terms.

10 days ago I started a thread in sci.math
"Maximal determinant of matrix with elements 1..n^2"
You can locate the thread in Google Groups by searching for
"pfoertner determinant" or read it at
http://mathforum.org/discuss/sci.math/t/539004

This is my last post to sci.math:
(had planned to submit it as A088238, but only one non-zero term
seems a little bit poor)
Hugo Pfoertner wrote:
> 
> Kent Paul Dolan wrote:
> >
> > "Hugo Pfoertner" <nothing at abouthugo.de> wrote;
> > > For n=7 I'm currently ... improved to
> > > a(7)=761973807872 for matrix
> >
> > > 37  6 47 35 18  9 23
> > > 49 30 12  3 26 19 36
> > > 25 40 33 21  1 43 11
> > >  2 20 39  8 34 31 41
> > > 17 22  5 46 13 27 44
> > > 28 10 15 29 45 42  7
> > > 16 48 24 32 38  4 14
> >
> > And yet since you suspect the largest terms should be on the main
> > diagonal, and only one of them is, it seems you have a very long way to
> > go indeed.
> >
> 
> [...]
> 
> > xanthian.
> 
> Using row permutations the matrix can also be written
> (did not check the sign of the determinant)
> 
> 49 30 12  3 26 19 36
> 16 48 24 32 38  4 14
> 37  6 47 35 18  9 23
> 17 22  5 46 13 27 44
> 28 10 15 29 45 42  7
> 25 40 33 21  1 43 11
>  2 20 39  8 34 31 41
> 
> which is not too far from the "max diagonal" form. My earlier
> conjecture that the optimal solutions might have the n largest
> elements on the diagonal does probably not hold for n>5
> because also my n=6 solution, which I couldn't improve has
> different diagonal elements in the lower right corner.
> 
> Hugo

In the meantime the OEIS entry
http://www.research.att.com/projects/OEIS?Anum=A085000 has
been updated twice. There are two minor glitches in the updated page:
The estimate given by Tim Paulden for a(6) was
"a(6) is at least 1862125166." and not 1865999570 (which is the value
I've found a little bit later with my program). The example given for
the
3X3 matrix says:
"The following 3 X 3 matrix is one of 72 whose determinant is 412:"
There are only 36 matrices with det(A_3)=412, the other 36 matrices have
det(A_3)=-412.
Currently I'm searching n=7 with best result found so far
det(A_7)=762074188050 with A_7=
49  5 14 19 23 24 42
25 48 33 15 10  7 37
21 18 47  4 26 43 16
 8  9 39 46 29 13 32
38 31 22 27 45 11  1
28 30 17 44  2 41 12
 6 34  3 20 40 36 35

To test formulas for the asymptotic behavior I've als run the
same program for n=8,9,10:
a(8)>=4.40799*10^14
a(9)>=3.46119*10^17
a(10)>=3.568545*10^20

The heuristic formula a(n)~=0.44 * n^(2.06*n) doesn't fit the data
for n>=6. Can we find a more appropriate formula?

An observation I made for the "optimal" matrices: All row and
column sums are very close or equal to the average value
n*(n^2+1)/2. For A_7 given above all row or column sums are
either 174,175 or 176.

There are closely related problems, that are even harder to solve.
See OEIS:
http://www.research.att.com/projects/OEIS?Anum=A088237
http://www.research.att.com/projects/OEIS?Anum=A088216
http://www.research.att.com/projects/OEIS?Anum=A088217

Maybe the toughest of all:
How many different of these matrices have determinant 0?
I had planned to submit it as A088238, but only one non-zero term
seems a little bit poor :-(

I only know that 0 2X2 matrices and 36*76=2736 3X3 have
det(A_n)=0. (See http://www.research.att.com/projects/OEIS?Anum=A088215
)

Hugo Pfoertner