A323298 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have exactly one vertex in common.

1, 0, 0, 1, 0, 15, 150, 1815, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0, 6190283353629375, 0, 191898783962510625, 0, 6332659870762850625, 0, 221643095476699771875

Offset

0,6

Comments

The only way to cover more than 7 vertices is with edges all having a single common vertex. For the special cases of n = 6 or n = 7, there are also covers without a common vertex. - Andrew Howroyd, Aug 15 2019

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

a(2*n) = 0 for n > 3; a(2*n-1) = A001147(n) for n > 4. - Andrew Howroyd, Aug 15 2019

Example

The a(5) = 15 hypergraphs:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}
The following are non-isomorphic representatives of the 5 unlabeled 3-uniform hypergraphs spanning 7 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 1815.
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
  630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
   30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
From Andrew Howroyd, Aug 15 2019: (Start)
The following are non-isomorphic representatives of the 2 unlabeled 3-uniform hypergraphs spanning 6 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(6) = 150.
    120 X {{1,2,3},{1,4,5},{3,5,6}}
     30 X {{1,2,3},{1,4,5},{3,5,6},{2,4,6}}
(End)

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}]

Prog

(PARI) a(n)={if(n%2, if(n<=3, n==3, if(n==7, 1815, n!/(2^(n\2)*(n\2)!))), if(n==6, 150, n==0))} \\ Andrew Howroyd, Aug 15 2019

Crossrefs

Cf. A001147, A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323299.

Sequence in context: A084902 A021364 A352160 * A206366 A016103 A206361

Adjacent sequences: A323295 A323296 A323297 * A323299 A323300 A323301

Keyword

nonn

Author

Gus Wiseman, Jan 11 2019

Extensions

Terms a(13) and beyond from Andrew Howroyd, Aug 15 2019

Status

approved