OFFSET
0,5
COMMENTS
Here, row span means the cardinality of the row space.
LINKS
M. Breen and D Hume, On the cardinalities of row spaces of Boolean matrices, Technical Report, no. 1999-3, Tennessee Technological University.
FORMULA
T(n,1) = 1 (the zero matrix).
T(n,2^n) = n! (the permutation matrices).
T(n,2) = (2^n-1)^2.
For k > 2^(n-1), T(n,k) is nonzero iff k=2^(n-1)+2^j for any j in {0,1,2,...,n-1}.
EXAMPLE
Triangle begins:
1;
1, 1;
1, 9, 4, 2;
1, 49, 144, 198, 78, 36, 0, 6;
1, 225, 2500, 9650, 15864, 17640, 8784, 6936, 2304, 1320, 0, 288, 0, 0, 0, 24;
...
T(2,3)=4 because we have: {{0, 1}, {1, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}, {{1, 1}, {1, 0}}.
MATHEMATICA
B[n_] := Tuples[Tuples[{0, 1}, n], n]; rowspace[matrix_, n_] := Sort[DeleteDuplicates[Clip[Tuples[{0, 1}, n].matrix]]]; Table[Table[
Count[Map[Length[rowspace[#, n]] &, B[n]], k], {k, 1, 2^n}], {n, 0, 4}] // Grid
CROSSREFS
KEYWORD
nonn,tabf,more
AUTHOR
Geoffrey Critzer, May 09 2023
STATUS
approved