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A261041 Number of partitions of subsets of {1,...,n}, where consecutive integers are required to be in different parts.

%I A261041 #44 Sep 10 2024 16:25:14
%S A261041 1,2,4,10,29,97,366,1534,7050,35167,188835,1084180,6618472,42756208,
%T A261041 291120551,2081922515,15590248868,121920095674,993343650912,
%U A261041 8414029179365,73953763887277,673316834487162,6340176007793060,61657373569634586,618445940056365121
%N A261041 Number of partitions of subsets of {1,...,n}, where consecutive integers are required to be in different parts.
%C A261041 From _Gus Wiseman_, Nov 25 2019: (Start)
%C A261041 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 A261041   {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
%C A261041          {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}
%C A261041                     {{1,2},{3}}    {{1,2},{3,4}}
%C A261041                     {{1},{2},{3}}  {{1,2,3},{4}}
%C A261041                                    {{1,4},{2,3}}
%C A261041                                    {{1},{2},{3,4}}
%C A261041                                    {{1},{2,3},{4}}
%C A261041                                    {{1,2},{3},{4}}
%C A261041                                    {{1,4},{2},{3}}
%C A261041                                    {{1},{2},{3},{4}}
%C A261041 (End)
%H A261041 Alois P. Heinz, <a href="/A261041/b261041.txt">Table of n, a(n) for n = 0..576</a>
%H A261041 Notamathematician et al., <a href="https://mathoverflow.net/q/478446">Generating function for A261041</a>, MathOverflow, 2024.
%F A261041 From _Max Alekseyev_, Sep 08 2024: (Start)
%F A261041 a(n) = Sum_{k=0..n} A000110(k) * Sum_{j=0..[(n-k)/2]} binomial(k+j-1,j).
%F A261041 G.f.: 1/(1-x) * Sum_{k>=0} A000110(k) * (x/(1-x^2))^k. (End)
%e A261041 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 A261041 From _Gus Wiseman_, Nov 25 2019: (Start)
%e A261041 The a(0) = 1 through a(3) = 10 set partitions:
%e A261041   {}  {}     {}         {}
%e A261041       {{1}}  {{1}}      {{1}}
%e A261041              {{2}}      {{2}}
%e A261041              {{1},{2}}  {{3}}
%e A261041                         {{1,3}}
%e A261041                         {{1},{2}}
%e A261041                         {{1},{3}}
%e A261041                         {{2},{3}}
%e A261041                         {{1,3},{2}}
%e A261041                         {{1},{2},{3}}
%e A261041 (End)
%p A261041 g:= proc(n, l, t) option remember; `if`(n=0, 1, add(`if`(l>0
%p A261041       and j=l, 0, g(n-1, j, `if`(j=t, t+1, t))), j=0..t))
%p A261041     end:
%p A261041 a:= n-> g(n, 0, 1):
%p A261041 seq(a(n), n=0..30);
%t A261041 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 A261041 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A261041 Table[Length[Select[Join@@sps/@Subsets[Range[n]],!MemberQ[#,{___,x_,y_,___}/;x+1==y]&]],{n,0,6}] (* _Gus Wiseman_, Nov 25 2019 *)
%o A261041 (PARI) a261041(n) = sum(k=0,n, sum(j=0,k,stirling(k,j,2)) * sum(j=0,(n-k)\2, binomial(k+j-1,j))); \\ _Max Alekseyev_, Sep 08 2024
%Y A261041 Cf. A000110, A003242, A114901, A247100, A261134, A261489, A261492, A273461, A274174.
%Y A261041 Partial sums of A376077.
%K A261041 nonn
%O A261041 0,2
%A A261041 _Alois P. Heinz_, Aug 09 2015