binary matrices with zero diagonal & no zero rows or columns

[Brendan McKay]

Folks, my posting was rejected by the seqfan server.  I don't know
if there is something wrong with the server in general or just with
respect to me.  If this one does not appear via the server, can one of
you send it?

Thanks, Brendan.

-----------------------------------------------------

* wouter meeussen <wouter.meeussen at pandora.be> [030827 05:55]:
> Mathematica and I agree with you:
> 
> Table[it = (Partition[#1, n] &) /@
>       IntegerDigits[Range[0, -1 + 2^n^2], 2, n^2];
>   Count[it, (q_)?MatrixQ /;
>       Tr[q] === 0 && (Times @@ (Plus @@@ q)) >
>           0 && (Times @@ (Plus @@@ Transpose[q]) > 0)], {n, 1, 4}]
> Out[]=
> {0, 1, 18, 1699}

I scribbled down an inclusion-exclusion for this, applied the binomial
theorem twice, and to my great amazement got perfect agreement:

  sum( f(n,r), r=0..n )   where

f(n,r)
 = binomial(n,r) (-1)^r (1-2^(-n+r+1))^(n-r) (1-2^(-n+r))^r 2^((n-r)(n-1))

Values:
  0, 1, 18, 1699, 592260, 754179301, 3562635108438, 63770601591579079,
        4405870283636411477640, 1190873924687350003735546441,
        1270602397076493907445608866890778,
        5381240610642043789096251476993474339179 

Brendan.

----- End forwarded message -----