Permanents and determinants of (0,1) matrices

[Edwin Clark]

Hugo wrote:

> I'm not quite sure if the same can be done for A089474 (integrating it
> into A052655). Would be one of the nice cases where an existing
> sequence has an unexpected application to another problem.

Hugo, Jaap, et al:

Note that for sequence A052655 except at n = 2, a(n) = n*n!. 

I think it is reasonable when you think about it that sequence

ID Number: A089474
URL:       http://www.research.att.com/projects/OEIS?Anum=A089474
Sequence:  1,6,18,96,600
Name:      Number of real non-singular {0,1}-matrices of order n having
maximal permanent = A089473(n).

SHOULD have a(n) = n*n! when n > 2. Reasoning is that the maximum
permanent is per A where A has all 1's except for n-1 0's on the main
diagonal. There are n ways to put the single 1 on the main diagonal and
then the columns are distinct so there are n! ways to permute them without
changing the permanent. That is, put n-1 0's in A so that no row or column
contains more than one 0. It remains to show that if you put more than n-1
0's in a binary matrix then either det = 0 or the permanent is decreased.

--Edwin