A247231 - OEIS

%I #34 Jul 06 2023 14:19:25

%S 1,1,3,1,9,19,1,21,114,219,1,45,475,2190,4231,1,93,1710,14235,63465,

%T 130023,1,189,5719,76650,592340,2730483,6129859,1,381,18354,372519,

%U 4442550,34586118,171636052,431723379,1,765,57475,1701630,29409681,344040858,2831994858,15542041644,44511042511

%N 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 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 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 Alois P. Heinz, <a href="/A247231/b247231.txt">Rows n = 1..18</a>

%F E.g.f.: A(y*(exp(x) - 1)) where A(x) is the e.g.f. for A001035.

%e Triangle T(n,k) begins:

%e   1;

%e   1,   3;

%e   1,   9,   19;

%e   1,  21,  114,   219;

%e   1,  45,  475,  2190,   4231;

%e   1,  93, 1710, 14235,  63465,  130023;

%e   1, 189, 5719, 76650, 592340, 2730483, 6129859;

%e   ...

%t A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]];

%t lg = Length[A001035];

%t A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];

%t 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 Row sums gives A000798, n >= 1.

%Y Leading diagonal gives A001035, n >= 1.

%Y Apparently column 2 gives the terms > 1 of A068156.

%K nonn,tabl

%O 1,3

%A _Geoffrey Critzer_, Nov 27 2014