powers of Antisymmetric Signed binary Matrices

[Edwin Clark]

On Sun, 24 Aug 2003, wouter meeussen wrote:

> superseeker came up blank for 2, 14, 92, 796 ...
> the number of antisymmetric (Tij = -Tji ;i<j) signed binary matrices of size n, whose powers T^k (k
> <= n^2) are binary signed matrices too.

Wouter:

I was getting different numbers, but then I finally noticed that in your
definition of antisymmetric you allow arbitrary diagonal elements. Perhaps
you should call your matrices quasi-antisymmetric or something like
that?

The usual definition of antisymmetric is Tij = -Tji for all i,j. This
means that the diagonal elements should be 0. What you have is of the form
D + T where D is diagonal and T is antisymmetric.

Edwin

> 
> It turns out those powers all are *antisymmetric* signed binary matrices.
> The a(n=4)= 796 matrices have T^k =Identity for k in {1, 2, 3, 4, 6 or 12},
> 1 times k=1
> 15 times k=2
> 72 times k=3
> 300 times k=4
> 312 times k=6
> 96 times k=12.
> 
> They have
> 76 times 4 non-zero elements,
> 288 times 5 non-zero elements,
> 192 times 6 non-zero elements,
> 192 times 8 non-zero elements,
> 48 times 12 non-zero elements.
> 
> The 5 by 5 matrices are kindo' hard to generate,
> scanning through -1+3^(5*(5+1)/2) = 14,348,906 candidates.
> Have it running now. Any takers to check my counts?
> 
> 
> Wouter.
> 
> 
> ************* sorry for the mess below  *****************
> Table[it =
>     triamatsig/@ (-1+IntegerDigits[Range[0,-1+3^(n(n+1)/2)],3,n(n+1)/2]);
>      Count[it,(q_?MatrixQ)/; Det[q]=!=0 && And@@ Table[
>             Union[Flatten[{MatrixPower[q,k],{-1,0,1}}]]==={-1,0,1},
>             {k, n^2}] ], {n, 4}]
> with
> triamatsig[li_List] :=
>   Block[{len = Sqrt[8*Length[li]+1]/2-1/2},
>     If[IntegerQ[len],
>       (Part[li,#]& /@ Table[If[j>i,j(j-1)/2+i,i(i-1)/2+j],{i,len},
>         {j,len}])Table[If[j>i,-1,1],{i,len},{j,len}],li]]
> 
> 
> 
> 
> 

------------------------------------------------------------
    W. Edwin Clark, Math Dept, University of South Florida,
           http://www.math.usf.edu/~eclark/
------------------------------------------------------------