A046908 -id:A046908 - cp's OEIS frontend

A354615 Triangular array read by rows: T(n,k) is the number of labeled posets on [n] that are composed of exactly k irreducible posets, n >= 0, 0 <= k <= n.

1, 0, 1, 0, 1, 2, 0, 7, 6, 6, 0, 97, 62, 36, 24, 0, 2251, 1110, 510, 240, 120, 0, 80821, 30902, 11340, 4440, 1800, 720, 0, 4305127, 1273566, 369726, 119280, 42000, 15120, 5040, 0, 332273257, 75831422, 17192196, 4476024, 1335600, 433440, 141120, 40320

Offset: 0

Geoffrey Critzer, Jul 08 2022

			  1;
  0,    1;
  0,    1,    2;
  0,    7,    6,   6;
  0,   97,   62,  36,  24;
  0, 2251, 1110, 510, 240, 120;
  ...

Cf. A046908 (column k=1), A001035 (row sums), A000142 (main diagonal).

nn = 9; A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt",
  "Table"], {, }][[All, 2]]*   Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]]; Table[Take[(Range[0, nn]!* CoefficientList[ Series[1/(1 - y (1 - 1/A[x])), {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}]

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

A046907 Number of isomorphism classes of irreducible posets with n labeled points.

Original entry on oeis.org

1, 1, 1, 2, 7, 31, 184, 1351, 12524, 146468, 2177570, 41374407, 1008220289, 31559446774, 1269310589336, 65562045668340, 4345161435996517

Offset: 0

John A. Wright

G. Brinkmann, B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table 1).
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

Cf. A046908.

A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]];
B[x_] = A000112.x^Range[0, Length[A000112] - 1];
A[x_] = 2 - 1/B[x];
CoefficientList[A[x] + O[x]^Length[A000112], x] (* Jean-François Alcover, Jan 01 2020 *)

G.f.: A(x) = 2-1/B(x), where B(x) is g.f. of A000112. - Vladeta Jovovic, Jan 15 2006

More terms from Vladeta Jovovic, Jan 15 2006

A352399 Triangular array read by rows: T(n,k) is the number of partial order relations on [n] that have exactly k components, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 12, 6, 1, 0, 146, 60, 12, 1, 0, 3060, 970, 180, 20, 1, 0, 101642, 24180, 3750, 420, 30, 1, 0, 5106612, 901334, 110040, 10990, 840, 42, 1, 0, 377403266, 49347228, 4567976, 376320, 27020, 1512, 56, 1, 0, 40299722580, 3923052354, 269812620, 17322648, 1071000, 58716, 2520, 72, 1

Offset: 0

Geoffrey Critzer, Jul 05 2022

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    2,   1;
  0,   12,   6,   1;
  0,  146,  60,  12,  1;
  0, 3060, 970, 180, 20, 1;
  ...

Cf. A001927 (column 1), A001035 (row sums), A046908.

nn = 8; A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt",
      "Table"], {, }][[All, 2]]* Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];
Table[Take[(Range[0, nn]! CoefficientList[Series[A[x]^y, {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}] // Grid

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

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A354615 Triangular array read by rows: T(n,k) is the number of labeled posets on [n] that are composed of exactly k irreducible posets, n >= 0, 0 <= k <= n.

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A046907 Number of isomorphism classes of irreducible posets with n labeled points.

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A352399 Triangular array read by rows: T(n,k) is the number of partial order relations on [n] that have exactly k components, n>=0, 0<=k<=n.

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