binary matrices with zero diagonal & no zero rows or columns

[Vladeta]

----- Original Message -----
From: "Brendan McKay" <bdm at cs.anu.edu.au>
To: "wouter meeussen" <wouter.meeussen at pandora.be>
Cc: "Edwin Clark" <eclark at math.usf.edu>; "seqfan" <seqfan at ext.jussieu.fr>
Sent: Wednesday, August 27, 2003 11:12 AM
Subject: Re: binary matrices with zero diagonal & no zero rows or columns


>
> 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 -----
>


Or

sum( (-1)^(n-r)*binomial(n,r)*(2^(r-1)-1)^r*(2^r-1)^(n-r), r=0..n ).


Vl.