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%I #35 Nov 27 2019 07:52:10
%S 1,2,4,10,29,97,366,1534,7050,35167,188835,1084180,6618472,42756208,
%T 291120551,2081922515,15590248868,121920095674,993343650912,
%U 8414029179365,73953763887277,673316834487162,6340176007793060,61657373569634586,618445940056365121
%N Number of partitions of subsets of {1,...,n}, where consecutive integers are required to be in different parts.
%C From _Gus Wiseman_, Nov 25 2019: (Start)
%C 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:
%C {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
%C {{1},{2}} {{1},{2,3}} {{1},{2,3,4}}
%C {{1,2},{3}} {{1,2},{3,4}}
%C {{1},{2},{3}} {{1,2,3},{4}}
%C {{1,4},{2,3}}
%C {{1},{2},{3,4}}
%C {{1},{2,3},{4}}
%C {{1,2},{3},{4}}
%C {{1,4},{2},{3}}
%C {{1},{2},{3},{4}}
%C (End)
%H Alois P. Heinz, <a href="/A261041/b261041.txt">Table of n, a(n) for n = 0..400</a>
%e For n=3 the a(3) = 10 partitions are {}, 1, 2, 3, 1|2, 13, 1|3, 2|3, 13|2, 1|2|3.
%e From _Gus Wiseman_, Nov 25 2019: (Start)
%e The a(0) = 1 through a(3) = 10 set partitions:
%e {} {} {} {}
%e {{1}} {{1}} {{1}}
%e {{2}} {{2}}
%e {{1},{2}} {{3}}
%e {{1,3}}
%e {{1},{2}}
%e {{1},{3}}
%e {{2},{3}}
%e {{1,3},{2}}
%e {{1},{2},{3}}
%e (End)
%p g:= proc(n, l, t) option remember; `if`(n=0, 1, add(`if`(l>0
%p and j=l, 0, g(n-1, j, `if`(j=t, t+1, t))), j=0..t))
%p end:
%p a:= n-> g(n, 0, 1):
%p seq(a(n), n=0..30);
%t 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 *)
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t Table[Length[Select[Join@@sps/@Subsets[Range[n]],!MemberQ[#,{___,x_,y_,___}/;x+1==y]&]],{n,0,6}] (* _Gus Wiseman_, Nov 25 2019 *)
%Y Cf. A003242, A114901, A247100, A261134, A261489, A261492, A273461, A274174.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Aug 09 2015
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