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A322064
Number of ways to choose a stable partition of a simple connected graph with n vertices.
7
1, 1, 1, 7, 141, 6533, 631875, 123430027, 48659732725, 39107797223409, 64702785181953175, 221636039917857648631, 1575528053913118966200441, 23249384407499950496231003021, 711653666389829384034090082068939, 45128328085994437067694854477617868995
OFFSET
0,4
COMMENTS
A stable partition of a graph is a set partition of the vertices where no non-singleton edge has both ends in the same block.
LINKS
EXAMPLE
The a(3) = 7 stable partitions. The simple connected graph is on top, and below is a list of all its stable partitions.
{1,3}{2,3} {1,2}{2,3} {1,2}{1,3} {1,2}{1,3}{2,3}
-------- -------- -------- --------
{{1,2},{3}} {{1,3},{2}} {{1},{2,3}} {{1},{2},{3}}
{{1},{2},{3}} {{1},{2},{3}} {{1},{2},{3}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Sum[Length[Select[Subsets[Complement[Subsets[Range[n], {2}], Union@@Subsets/@stn]], And[Union@@#==Range[n], Length[csm[#]]==1]&]], {stn, sps[Range[n]]}], {n, 5}]
PROG
(PARI) \\ See A322278 for M.
seq(n)={concat([1], (M(n)*vectorv(n, i, 1))~)} \\ Andrew Howroyd, Dec 01 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 25 2018
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Dec 01 2018
STATUS
approved