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.

1, 1, 3, 1, 9, 19, 1, 21, 114, 219, 1, 45, 475, 2190, 4231, 1, 93, 1710, 14235, 63465, 130023, 1, 189, 5719, 76650, 592340, 2730483, 6129859, 1, 381, 18354, 372519, 4442550, 34586118, 171636052, 431723379, 1, 765, 57475, 1701630, 29409681, 344040858, 2831994858, 15542041644, 44511042511

Offset: 1

Author

Geoffrey Critzer, Nov 27 2014

Keywords

• nonn
• tabl

Comments

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

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

Examples

			Triangle T(n,k) begins:
  1;
  1,   3;
  1,   9,   19;
  1,  21,  114,   219;
  1,  45,  475,  2190,   4231;
  1,  93, 1710, 14235,  63465,  130023;
  1, 189, 5719, 76650, 592340, 2730483, 6129859;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..18

Crossrefs

Row sums gives A000798, n >= 1.

Leading diagonal gives A001035, n >= 1.

Apparently column 2 gives the terms > 1 of A068156.

Programs

Mathematica

A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {, }][[All, 2]];
lg = Length[A001035];
A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
Rest[CoefficientList[#, y]]& /@ (CoefficientList[A[y*(Exp[x] - 1)] + O[x]^lg, x]*Range[0, lg - 1]!) // Flatten (* Jean-François Alcover, Jan 01 2020 *)

Formula

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