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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-Fran├žois 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.)