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.

Showing 1-4 of 4 results.

A323293 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 271, 5596, 231577, 21286940, 4392750641, 2100400533176
Offset: 0

Views

Author

Gus Wiseman, Jan 10 2019

Keywords

Examples

			The a(5) = 26 hypergraphs:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,2,5}}
  {{1,3,4}}
  {{1,3,5}}
  {{1,4,5}}
  {{2,3,4}}
  {{2,3,5}}
  {{2,4,5}}
  {{3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
Non-isomorphic representatives of the 6 unlabeled 3-uniform hypertrees spanning 6 vertices where no two edges have two vertices in common, and their multiplicities in the labeled case which add up to a(6) = 271:
    1 X {}
   20 X {{1,2,3}}
   90 X {{1,2,5},{3,4,5}}
   10 X {{1,2,3},{4,5,6}}
  120 X {{1,3,5},{2,3,6},{4,5,6}}
   30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}

Crossrefs

Essentially the same as A287232.
Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>1&]],{n,8}]

Extensions

a(9) from Andrew Howroyd, Aug 14 2019
a(10) and a(11) (using A287232) from Joerg Arndt, Oct 12 2023

A323294 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have two vertices in common.

Original entry on oeis.org

1, 0, 0, 1, 11, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431
Offset: 0

Views

Author

Gus Wiseman, Jan 10 2019

Keywords

Examples

			The a(4) = 11 hypergraphs:
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Links

Crossrefs

Cf. A000217, A000665, A025035, A125791, A161680, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]<=1&],Union@@#==Range[n]&]],{n,10}]
  • PARI
    seq(n)={Vec(serlaplace(1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2))} \\ Andrew Howroyd, Aug 18 2019

Formula

a(n) = binomial(n,2) for n >= 5. - Gus Wiseman, Jan 16 2019
Binomial transform is A289837. - Gus Wiseman, Jan 16 2019
a(n) = A000217(n-1) for n >= 5. - Alois P. Heinz, Jan 24 2019
E.g.f.: 1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2. - Andrew Howroyd, Aug 18 2019

A003190 Number of connected 2-plexes.

Original entry on oeis.org

1, 0, 1, 3, 29, 2101, 7011181, 1788775603301, 53304526022885278403, 366299663378889804782330207902, 1171638318502622784366970315262493034215728, 3517726593606524901243694560022510194169866584119717555335
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The Palmer reference (incorrectly) has a(7)=7011349, a(8)=1788775603133, a(9)=53304526022885278659. - Sean A. Irvine, Mar 05 2015
Also connected 3-uniform hypergraphs on n vertices. - Gus Wiseman, Feb 23 2019

Examples

			From _Gus Wiseman_, Feb 23 2019: (Start)
Non-isomorphic representatives of the a(5) = 29 2-plexes:
  {{125}{345}}
  {{123}{245}{345}}
  {{135}{245}{345}}
  {{145}{245}{345}}
  {{123}{145}{245}{345}}
  {{124}{135}{245}{345}}
  {{125}{135}{245}{345}}
  {{134}{235}{245}{345}}
  {{145}{235}{245}{345}}
  {{123}{124}{135}{245}{345}}
  {{123}{145}{235}{245}{345}}
  {{124}{134}{235}{245}{345}}
  {{134}{145}{235}{245}{345}}
  {{135}{145}{235}{245}{345}}
  {{145}{234}{235}{245}{345}}
  {{123}{124}{134}{235}{245}{345}}
  {{123}{134}{145}{235}{245}{345}}
  {{123}{145}{234}{235}{245}{345}}
  {{124}{135}{145}{235}{245}{345}}
  {{125}{135}{145}{235}{245}{345}}
  {{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{235}{245}{345}}
  {{124}{135}{145}{234}{235}{245}{345}}
  {{125}{135}{145}{234}{235}{245}{345}}
  {{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{234}{235}{245}{345}}
  {{125}{134}{135}{145}{234}{235}{245}{345}}
  {{124}{125}{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{125}{134}{135}{145}{234}{235}{245}{345}}
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=3 of A301924.
Cf. A000665 (unlabeled 3-uniform), A025035, A125791 (labeled 3-uniform), A289837, A301922, A302374 (labeled 3-uniform spanning), A302394, A306017, A319540, A320395, A322451 (unlabeled 3-uniform spanning), A323292-A323299.

Formula

Inverse Euler transform of A000665. - Sean A. Irvine, Mar 05 2015

Extensions

a(7)-a(9) corrected and extended by Sean A. Irvine, Mar 05 2015

A320446 Covers of triangles by tetrahedra: number of labeled 4-uniform hypergraphs spanning n vertices such that every three vertices appear together in some edge.

Original entry on oeis.org

1, 1, 1, 0, 1, 6, 5789
Offset: 0

Views

Author

Gus Wiseman, Jan 10 2019

Keywords

Examples

			The a(5) = 6 hypergraphs:
  {{1234},{1235},{1245},{1345}}
  {{1234},{1235},{1245},{2345}}
  {{1234},{1235},{1345},{2345}}
  {{1234},{1245},{1345},{2345}}
  {{1235},{1245},{1345},{2345}}
  {{1234},{1235},{1245},{1345},{2345}}

Crossrefs

Cf. A006126, A038041, A299471, A301922, A302374, A302394, A306021, A319540, A320395, A322451.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{4}]],Length[Union@@(Subsets[#,{3}]&/@#)]==Binomial[n,3]&]],{n,6}]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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