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A326249 Number of capturing set partitions of {1..n} that are not nesting.

%I A326249 #12 Oct 29 2024 19:54:10
%S A326249 0,0,0,0,0,1,9,55,283,1324,5838,24744
%N A326249 Number of capturing set partitions of {1..n} that are not nesting.
%C A326249 Capturing is a weaker condition than nesting. A set partition is capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. For example, {{1,3,5},{2,4}} is capturing but not nesting, so is counted under a(5).
%e A326249 The a(6) = 9 set partitions:
%e A326249   {{1},{2,4,6},{3,5}}
%e A326249   {{1,3,5},{2,4},{6}}
%e A326249   {{1,3,6},{2,4},{5}}
%e A326249   {{1,3,6},{2,5},{4}}
%e A326249   {{1,4,6},{2},{3,5}}
%e A326249   {{1,4,6},{2,5},{3}}
%e A326249   {{1,3,5},{2,4,6}}
%e A326249   {{1,2,4,6},{3,5}}
%e A326249   {{1,3,5,6},{2,4}}
%t A326249 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A326249 capXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<t<y||z<x<y<t];
%t A326249 nesXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;x<z<t<y||z<x<y<t];
%t A326249 Table[Length[Select[sps[Range[n]],!nesXQ[#]&&capXQ[#]&]],{n,0,8}]
%Y A326249 MM-numbers of capturing, non-nesting multiset partitions are A326260.
%Y A326249 Nesting set partitions are A016098.
%Y A326249 Capturing set partitions are A326243.
%Y A326249 Non-crossing, nesting set partitions are A122880 (conjectured).
%Y A326249 Cf. A000110, A054391, A058681, A095661, A099947.
%Y A326249 Cf. A326212, A326237, A326245, A326246, A054391, A326255, A326256.
%K A326249 nonn,more
%O A326249 0,7
%A A326249 _Gus Wiseman_, Jun 20 2019