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

0,3

Comments

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

Links

Table of n, a(n) for n=0..44.

Geoffrey Critzer, Finite Topologies

Formula

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

Example

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;
  ...

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 *)

Crossrefs

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

Sequence in context: A189963 A156649 A197330 * A168229 A019693 A007493

Adjacent sequences: A343879 A343880 A343881 * A343883 A343884 A343885

Keyword

nonn,tabl

Author

Geoffrey Critzer, May 02 2021

Status

approved