cp's OEIS Frontend

A322064 Number of ways to choose a stable partition of a simple connected graph with n vertices.

%I A322064 #10 Dec 01 2018 21:38:02
%S A322064 1,1,1,7,141,6533,631875,123430027,48659732725,39107797223409,
%T A322064 64702785181953175,221636039917857648631,1575528053913118966200441,
%U A322064 23249384407499950496231003021,711653666389829384034090082068939,45128328085994437067694854477617868995
%N A322064 Number of ways to choose a stable partition of a simple connected graph with n vertices.
%C A322064 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.
%H A322064 Andrew Howroyd, <a href="/A322064/b322064.txt">Table of n, a(n) for n = 0..75</a>
%e A322064 The a(3) = 7 stable partitions. The simple connected graph is on top, and below is a list of all its stable partitions.
%e A322064   {1,3}{2,3}     {1,2}{2,3}     {1,2}{1,3}     {1,2}{1,3}{2,3}
%e A322064   --------       --------       --------       --------
%e A322064   {{1,2},{3}}    {{1,3},{2}}    {{1},{2,3}}    {{1},{2},{3}}
%e A322064   {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}
%t A322064 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A322064 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]]]]]]]]];
%t A322064 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}]
%o A322064 (PARI) \\ See A322278 for M.
%o A322064 seq(n)={concat([1], (M(n)*vectorv(n,i,1))~)} \\ _Andrew Howroyd_, Dec 01 2018
%Y A322064 Row sums of A322278.
%Y A322064 Cf. A000110, A000569, A001187, A006125, A048143, A229048, A240936, A245883, A277203, A321911, A321979, A322063, A322065.
%K A322064 nonn
%O A322064 0,4
%A A322064 _Gus Wiseman_, Nov 25 2018
%E A322064 Terms a(7) and beyond from _Andrew Howroyd_, Dec 01 2018