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A326246 Number of crossing, capturing set partitions of {1..n}.

%I A326246 #7 Oct 29 2024 19:53:53
%S A326246 0,0,0,0,0,3,37,307,2173,14344,92402,596688
%N A326246 Number of crossing, capturing set partitions of {1..n}.
%C A326246 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 A326246 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 A326246 The a(5) = 3 set partitions:
%e A326246   {{1,3,4},{2,5}}
%e A326246   {{1,3,5},{2,4}}
%e A326246   {{1,4},{2,3,5}}
%t A326246 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A326246 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A326246 capXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<t<y||z<x<y<t];
%t A326246 Table[Length[Select[sps[Range[n]],capXQ[#]&&croXQ[#]&]],{n,0,5}]
%Y A326246 MM-numbers of crossing, capturing multiset partitions are A326259.
%Y A326246 Crossing set partitions are A016098.
%Y A326246 Capturing set partitions are A326243.
%Y A326246 Crossing, nesting set partitions are A326248.
%Y A326246 Crossing, non-capturing set partitions are A326245.
%Y A326246 Non-crossing, capturing set partitions are A122880 (conjecture).
%Y A326246 Cf. A000108, A000110, A058681, A099947, A324170, A326211, A326249, A054391, A326255.
%K A326246 nonn,more
%O A326246 0,6
%A A326246 _Gus Wiseman_, Jun 20 2019