binary matrices with zero diagonal & no zero rows or columns

[wouter meeussen]

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}

go get'm Edwin!

W.

----- Original Message -----
From: "Edwin Clark" <eclark at math.usf.edu>
To: "seqfan" <seqfan at ext.jussieu.fr>
Sent: Tuesday, August 26, 2003 8:54 PM
Subject: binary matrices with zero diagonal & no zero rows or columns


>
> Let a(n) be the number of n x n matrices with entries in {0,1} with no
> zero row and no zero column, AND with zero (main) diagonal.
>
> I get
>
> a(1) = 0, a(2) = 1, a(3) = 18, a(4) = 1699.
>
> I find it hard to believe that this is not in the OEIS, so perhaps this is
> wrong. I would appreciate it if someone would check these numbers.
>
> a(n) is also the number of simple labeled digraphs on n nodes for which
> every vertex has indegree at least one and outdegree at least one. This
> question was just raised on sci.math.research (if I interpret the question
> correctly).
>
> Edwin
>
>
>
>
>