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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.

%I A354741 #44 Jul 14 2022 08:57:40
%S A354741 1,1,1,1,9,6,1,49,306,156,1,225,8550,37488,19272,1,961,194850,4811700,
%T A354741 17551800,10995120
%N 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.
%C A354741 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.
%H A354741 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boolean_matrix">Boolean matrix</a>
%F A354741 T(n,0) = 1.
%F A354741 T(n,1) = (2^n-1)^2.
%F A354741 T(n,2) = (3^n - 2*2^n + 1)^2 + (1/2)*(4^n - 2*3^n + 2^n)^2.
%e A354741 Table begins:
%e A354741   1;
%e A354741   1,   1;
%e A354741   1,   9,    6;
%e A354741   1,  49,  306,   156;
%e A354741   1, 225, 8550, 37488, 19272;
%e A354741   ...
%t A354741 Table[B = Tuples[Tuples[{0, 1}, nn],nn]; bospan[matrix_]:= Sort[DeleteDuplicates[
%t A354741      Map[Clip[Total[#]] &, Drop[Subsets[matrix], 1]]]]; rowrank[matrix_] :=
%t A354741    If[Total[Map[Total, matrix]] == 0, 0, Length[Select[Drop[Subsets[DeleteCases[matrix, Table[0, {nn}]]], 1],
%t A354741        bospan[#] == bospan[DeleteCases[matrix, Table[0, {nn}]]] &][[ 1]]]]; Tally[
%t A354741     Table[rowrank[B[[i]]], {i, 1, 2^(nn^2)}]][[All,2]], {nn, 0, 4}] // Grid
%Y A354741 Columns k = 0 and 1 give A000012, A060867.
%Y A354741 Row sums give A002416.
%Y A354741 Cf. A048651, A064230, A286331, A355333.
%K A354741 nonn,tabl,more
%O A354741 0,5
%A A354741 _Geoffrey Critzer_, Jun 12 2022
%E A354741 Row n=5 from _Pontus von Brömssen_, Jul 14 2022