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A326245 Number of crossing, non-capturing set partitions of {1..n}.

%I A326245 #7 Oct 29 2024 19:54:31
%S A326245 0,0,0,0,1,7,34,141,537,1941,6777,23096,77340
%N A326245 Number of crossing, non-capturing set partitions of {1..n}.
%C A326245 A set partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y, and capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.
%H A326245 Eric Marberg, <a href="http://arxiv.org/abs/1203.5738">Crossings and nestings in colored set partitions</a>, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
%e A326245 The a(4) = 1 and a(5) = 7 set partitions:
%e A326245   {{1,3},{2,4}}  {{1,2,4},{3,5}}
%e A326245                  {{1,3},{2,4,5}}
%e A326245                  {{1},{2,4},{3,5}}
%e A326245                  {{1,3},{2,4},{5}}
%e A326245                  {{1,3},{2,5},{4}}
%e A326245                  {{1,4},{2},{3,5}}
%e A326245                  {{1,4},{2,5},{3}}
%t A326245 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A326245 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A326245 capXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<t<y||z<x<y<t];
%t A326245 Table[Length[Select[sps[Range[n]],!capXQ[#]&&croXQ[#]&]],{n,0,10}]
%Y A326245 Crossing set partitions are A016098.
%Y A326245 Non-capturing set partitions are A054391.
%Y A326245 Crossing, capturing set partitions are A326246.
%Y A326245 Cf. A000108, A000110, A001519, A054391, A058681, A095661, A324170.
%Y A326245 Cf. A326212, A326237, A326243, A326248, A326249.
%K A326245 nonn,more
%O A326245 0,6
%A A326245 _Gus Wiseman_, Jun 20 2019