A354741 Triangular array read by rows. T(n,k) is the number of n X n Boolean matrices with row rank k, n >= 0, 0 <= k <= n.

1, 1, 1, 1, 9, 6, 1, 49, 306, 156, 1, 225, 8550, 37488, 19272, 1, 961, 194850, 4811700, 17551800, 10995120

Offset: 0

Author

Geoffrey Critzer, Jun 12 2022

Keywords

• nonn
• tabl
• more

Comments

Compare to A286331 which counts n X n matrices over the field GF(2). Note that the limit when n->oo of the probability that a matrix over GF(2) has rank n is equal to Product_{i>=1} (1-1/2^i) = 0.288... (see A048651). Here, it appears (from some empirical computations) that the limiting probability that a Boolean matrix has rank n is 1.

Examples

			Table begins:
  1;
  1,   1;
  1,   9,    6;
  1,  49,  306,   156;
  1, 225, 8550, 37488, 19272;
  ...
		

Links

• Wikipedia, Boolean matrix

Crossrefs

Columns k = 0 and 1 give A000012, A060867.

Row sums give A002416.

Cf. A048651, A064230, A286331, A355333.

Programs

Mathematica

Table[B = Tuples[Tuples[{0, 1}, nn],nn]; bospan[matrix_]:= Sort[DeleteDuplicates[
     Map[Clip[Total[#]] &, Drop[Subsets[matrix], 1]]]]; rowrank[matrix_] :=
   If[Total[Map[Total, matrix]] == 0, 0, Length[Select[Drop[Subsets[DeleteCases[matrix, Table[0, {nn}]]], 1],
       bospan[#] == bospan[DeleteCases[matrix, Table[0, {nn}]]] &][[ 1]]]]; Tally[
    Table[rowrank[B[[i]]], {i, 1, 2^(nn^2)}]][[All,2]], {nn, 0, 4}] // Grid

Formula

T(n,0) = 1.

T(n,1) = (2^n-1)^2.

T(n,2) = (3^n - 2*2^n + 1)^2 + (1/2)*(4^n - 2*3^n + 2^n)^2.

Extensions

Row n=5 from Pontus von Brömssen, Jul 14 2022