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 A261041 Number of partitions of subsets of {1,...,n}, where consecutive integers are required to be in different parts. 9
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Gus Wiseman, Nov 25 2019: (Start) 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) LINKS Alois P. Heinz, Table of n, a(n) for n = 0..400 EXAMPLE For n=3 the a(3) = 10 partitions are {}, 1, 2, 3, 1|2, 13, 1|3, 2|3, 13|2, 1|2|3. From Gus Wiseman, Nov 25 2019: (Start) 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) MAPLE 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); MATHEMATICA 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 *) CROSSREFS Cf. A003242, A114901, A247100, A261134, A261489, A261492, A273461, A274174. Sequence in context: A320903 A230957 A279552 * A047051 A271077 A126349 Adjacent sequences:  A261038 A261039 A261040 * A261042 A261043 A261044 KEYWORD nonn AUTHOR Alois P. Heinz, Aug 09 2015 STATUS approved

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Last modified March 29 02:19 EDT 2020. Contains 333104 sequences. (Running on oeis4.)