Permanents and determinants of (0,1) matrices

[Gordon Royle]

> In the reference, sequence A000409 is given and there Neil writes :
>
> "What exactly is the difference between this sequence and A000410? -
>    njas" .
>
> Does someebody know ?


I can't figure it out exactly, but there is something seriously wrong with
the sequence definition... it is certainly *not* the number of singular n x
n (0,1)-matrices under any reasonable definition. For example, there are
definitely more than zero singular 2x2 (0,1)-matrices.

So somewhere there is an unstated condition on the matrices that are being
counted - we are only counting the singular matrices in a specific subclass
of all (0,1)-matrices. Checking the Metropolis and Stein paper, they
provided a lower bound for the class of singular (0,1)-matrices with
distinct non-zero rows..... in other words, if a matrix is "obviously"
singular because it has a zero row or two identical rows, then it isn't
worth counting...   this at least explains the 0 for n=2.

But it doesn't explain the 6 for n=3. And in fact, if we impose the
seemingly more natural condition that we must have distinct non-zero rows
AND columns, then the number is zero for n=3.

I think we need to ask Neil exactly what this sequence is *meant* to be
counting, and get the definition cleared up..