A343882 Triangular array read by rows: T(n,k) is the number of transitive relations on n labeled nodes with exactly k connected components.

1, 0, 2, 0, 9, 4, 0, 109, 54, 8, 0, 2647, 1115, 216, 16, 0, 110481, 36280, 6790, 720, 32, 0, 7291543, 1801927, 287475, 32020, 2160, 64, 0, 726434549, 133060816, 16873619, 1718290, 129080, 6048, 128, 0, 106312974249, 14380028959, 1387285830, 118346473, 8584240, 467488, 16128, 256

Offset: 0

Author

Geoffrey Critzer, May 02 2021

Keywords

• nonn
• tabl

Comments

T(n,n) = 2^n since each node is reflexive or not.

Examples

			Triangular array T(n,k) begins:
  1;
  0,      2;
  0,      9,     4;
  0,    109,    54,    8;
  0,   2647,  1115,  216,  16;
  0, 110481, 36280, 6790, 720, 32;
  ...
		

Links

• Geoffrey Critzer, Finite Topologies

Crossrefs

Cf. A245731 (column k=1), A006905 (row sums), A001035.

Programs

Mathematica

A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {, }][[All, 2]]* Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]]; nn = 10;
Range[0, nn]! CoefficientList[ Series[Exp[y Log[A[ x + Exp[ x] - 1]]], {x, 0, nn}], {x,y}] // Grid;Table[Take[(Range[0, nn]! CoefficientList[Series[Exp[y Log[A[ x + Exp[ x] - 1]]], {x, 0, nn}], {x, y}])[[i, All]], i], {i, 1, nn}] // Grid
  (* Import function in code after Jean-François Alcover *)

Formula

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