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A247231 Triangular array read by rows: T(n,k) is the number of ways to partition an n-set into exactly k blocks and then partially order the blocks, n>=1, 1<=k<=n.

%I A247231 #34 Jul 06 2023 14:19:25
%S A247231 1,1,3,1,9,19,1,21,114,219,1,45,475,2190,4231,1,93,1710,14235,63465,
%T A247231 130023,1,189,5719,76650,592340,2730483,6129859,1,381,18354,372519,
%U A247231 4442550,34586118,171636052,431723379,1,765,57475,1701630,29409681,344040858,2831994858,15542041644,44511042511
%N A247231 Triangular array read by rows: T(n,k) is the number of ways to partition an n-set into exactly k blocks and then partially order the blocks, n>=1, 1<=k<=n.
%C A247231 T(n,k) is also the number of topologies U on an n-set such that a minimal basis for U contains exactly k sets. - _Geoffrey Critzer_, Dec 26 2016
%C A247231 T(n,k) is also the number of transitive, reflexive Boolean relation matrices on [n] that have exactly k strongly connected components. - _Geoffrey Critzer_, Feb 27 2023
%H A247231 Alois P. Heinz, <a href="/A247231/b247231.txt">Rows n = 1..18</a>
%F A247231 E.g.f.: A(y*(exp(x) - 1)) where A(x) is the e.g.f. for A001035.
%e A247231 Triangle T(n,k) begins:
%e A247231   1;
%e A247231   1,   3;
%e A247231   1,   9,   19;
%e A247231   1,  21,  114,   219;
%e A247231   1,  45,  475,  2190,   4231;
%e A247231   1,  93, 1710, 14235,  63465,  130023;
%e A247231   1, 189, 5719, 76650, 592340, 2730483, 6129859;
%e A247231   ...
%t A247231 A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]];
%t A247231 lg = Length[A001035];
%t A247231 A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
%t A247231 Rest[CoefficientList[#, y]]& /@ (CoefficientList[A[y*(Exp[x] - 1)] + O[x]^lg, x]*Range[0, lg - 1]!) // Flatten (* _Jean-François Alcover_, Jan 01 2020 *)
%Y A247231 Row sums gives A000798, n >= 1.
%Y A247231 Leading diagonal gives A001035, n >= 1.
%Y A247231 Apparently column 2 gives the terms > 1 of A068156.
%K A247231 nonn,tabl
%O A247231 1,3
%A A247231 _Geoffrey Critzer_, Nov 27 2014