powers of Antisymmetric Signed binary Matrices

[wouter meeussen]

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.

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]]