A261041 - OEIS

%I #44 Sep 10 2024 16:25:14

%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..576</a>

%H Notamathematician et al., <a href="https://mathoverflow.net/q/478446">Generating function for A261041</a>, MathOverflow, 2024.

%F From _Max Alekseyev_, Sep 08 2024: (Start)

%F a(n) = Sum_{k=0..n} A000110(k) * Sum_{j=0..[(n-k)/2]} binomial(k+j-1,j).

%F G.f.: 1/(1-x) * Sum_{k>=0} A000110(k) * (x/(1-x^2))^k. (End)

%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 *)

%o (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 Cf. A000110, A003242, A114901, A247100, A261134, A261489, A261492, A273461, A274174.

%Y Partial sums of A376077.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Aug 09 2015