cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A342501 T(n,k) is the number of connected labeled posets with n elements and rank k: triangle read by rows.

Original entry on oeis.org

1, 0, 2, 0, 6, 6, 0, 38, 84, 24, 0, 390, 1710, 840, 120, 0, 6062, 49740, 36840, 8280, 720, 0, 134526, 2050566, 2184000, 646800, 85680, 5040, 0, 4172198, 118645044, 177549624, 65313360, 10735200, 947520, 40320
Offset: 1

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Author

R. J. Mathar, Mar 14 2021

Keywords

Comments

This is a variant of A342587 admitting only connected posets.

Examples

			The table starts in row n=1 and shows ranks k>=0:
1: 1
2: 0 2
3: 0 6 6
4: 0 38 84 24
5: 0 390 1710 840 120
6: 0 6062 49740 36840 8280 720
7: 0 134526 2050566 2184000 646800 85680 5040
8: 0 4172198 118645044 177549624 65313360 10735200 947520 40320

Links

Crossrefs

Cf. A001927 (row sums), A000142 (diagonal), A002031/A002027 (rank 1), A342500 (unlabeled).

A066303 Number of connected reduced partially ordered sets (posets) with n labeled elements.

Original entry on oeis.org

1, 0, 6, 50, 1220, 42122, 2278248, 182111666, 21094774212, 3479970392642, 807001730170592, 260359119927269882, 115887302333840512956, 70677462143507496419690, 58725460618979611312258632
Offset: 1

Views

Author

Christian G. Bower, Dec 12 2001

Keywords

Links

Formula

E.g.f. A(x)=B(x/(1+x)) where B(x) is e.g.f. of A001927.
E.g.f. A(x)=log(B(x)) where B(x) is e.g.f. of A066302.

A342588 T(n,k) is the number of labeled connected posets of n labeled elements with k covering relations (n>=1, k>=0). Triangle read by rows.

Original entry on oeis.org

1, 0, 2, 0, 0, 12, 0, 0, 0, 128, 18, 0, 0, 0, 0, 2000, 960, 100, 0, 0, 0, 0, 0, 41472, 43320, 15000, 1710, 140, 0, 0, 0, 0, 0, 0, 1075648, 1985760, 1453200, 490560, 90594, 10080, 770, 0, 0, 0, 0, 0, 0, 0, 33554432, 96937680, 122360000, 82220880, 32527488, 8205288, 1396640, 179760, 20048, 1050
Offset: 1

Views

Author

R. J. Mathar, Mar 16 2021

Keywords

Examples

			There are 8 connected unlabeled Hasse diagrams on 4 nodes with 3 arcs. 4 of them have automorphism group order 1, 2 of them have automorphism group order 2 and 2 have order 6. So T(4,3) = 4*4!/1 + 2*4!/2 + 2*4!/6 = 128.
There are 2 connected unlabeled Hasse diagrams on 4 nodes with 4 arcs, one has automorphism group order 2, the other 4. So T(4,4) = 1*4!/2+1*4!/4 = 18.
The triangle starts
1: 1
2: 0 2
3: 0 0 12
4: 0 0  0 128   18
5: 0 0  0   0 2000   960     100
6: 0 0  0   0    0 41472   43320   15000    1710    140
7: 0 0  0   0    0     0 1075648 1985760 1453200 490560 90594 10080 770

Links

Crossrefs

Cf. A001927 (row sums), A342589 (not necessarily connected), A342590 (unlabeled).

A046906 Number of connected irreducible posets with n labeled points.

Original entry on oeis.org

1, 1, 0, 0, 24, 1080, 52440, 3281880, 277953144, 32418855000, 5239070305080, 1173944480658840, 363936227764858584, 155521768202208047640, 91218870039317505477720, 73113879800794757415243480, 79743817918540500914682249144, 117883366412734188786535902826200, 235329353612778837110901775412557560
Offset: 0

Views

Author

John A. Wright

Keywords

References

  • J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.

Links

Crossrefs

A003431 gives isomorphism classes of these posets.

Programs

  • Mathematica
    nn = 18; A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt",
      "Table"], {, }][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];
Range[0, nn]! CoefficientList[ Series[(1 + Log[A[x]]) - A[ x] (1 - 1/A[x])^2 , {x, 0, nn}], x] (* Geoffrey Critzer, Jul 09 2022 *)

Formula

From Geoffrey Critzer, Jul 09 2022: (Start)
E.g.f.: 1 + log(A(x)) - A(x)(1-1/A(x))^2 where A(x) is the e.g.f. for A001035.
a(n) = A001927(n) - Sum_{k>=2} A354615(n,k). (End)

Extensions

a(8)-a(18) from Geoffrey Critzer, Jul 09 2022
a(0) changed to 1 by Geoffrey Critzer, Jul 10 2022

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

Views

Author

Geoffrey Critzer, Jul 05 2022

Keywords

Examples

			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;
  ...

Crossrefs

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

Programs

  • Mathematica
    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

Formula

E.g.f.: A(x)^y where A(x) is the e.g.f. for A001035.
Previous Showing 11-15 of 15 results.

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