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A261041
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Number of partitions of subsets of {1,...,n}, where consecutive integers are required to be in different parts.
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9
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1, 2, 4, 10, 29, 97, 366, 1534, 7050, 35167, 188835, 1084180, 6618472, 42756208, 291120551, 2081922515, 15590248868, 121920095674, 993343650912, 8414029179365, 73953763887277, 673316834487162, 6340176007793060, 61657373569634586, 618445940056365121
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OFFSET
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0,2
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COMMENTS
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Conjecture: Also the number of set partitions of {1, ..., n + 1} where, if x and x + 2 belong to the same block, then so does x + 1. For example, the a(0) = 1 through a(3) = 10 set partitions are:
{{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
{{1},{2}} {{1},{2,3}} {{1},{2,3,4}}
{{1,2},{3}} {{1,2},{3,4}}
{{1},{2},{3}} {{1,2,3},{4}}
{{1,4},{2,3}}
{{1},{2},{3,4}}
{{1},{2,3},{4}}
{{1,2},{3},{4}}
{{1,4},{2},{3}}
{{1},{2},{3},{4}}
(End)
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LINKS
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EXAMPLE
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For n=3 the a(3) = 10 partitions are {}, 1, 2, 3, 1|2, 13, 1|3, 2|3, 13|2, 1|2|3.
The a(0) = 1 through a(3) = 10 set partitions:
{} {} {} {}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{1},{2}} {{3}}
{{1,3}}
{{1},{2}}
{{1},{3}}
{{2},{3}}
{{1,3},{2}}
{{1},{2},{3}}
(End)
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MAPLE
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g:= proc(n, l, t) option remember; `if`(n=0, 1, add(`if`(l>0
and j=l, 0, g(n-1, j, `if`(j=t, t+1, t))), j=0..t))
end:
a:= n-> g(n, 0, 1):
seq(a(n), n=0..30);
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MATHEMATICA
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g[n_, l_, t_] := g[n, l, t] = If[n==0, 1, Sum[If[l>0 && j==l, 0, g[n-1, j, If[j==t, t+1, t]]], {j, 0, t}]]; a[n_] := g[n, 0, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 04 2017, translated from Maple *)
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[Join@@sps/@Subsets[Range[n]], !MemberQ[#, {___, x_, y_, ___}/; x+1==y]&]], {n, 0, 6}] (* Gus Wiseman, Nov 25 2019 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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