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%I #4 Jun 21 2019 22:45:14
%S 0,0,0,0,0,2,28,252,1890,13020,86564
%N Number of crossing, nesting set partitions of {1..n}.
%C 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 nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t.
%e The a(5) = 2 set partitions:
%e {{1,4},{2,3,5}}
%e {{1,3,4},{2,5}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t croXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;x<z<y<t||z<x<t<y];
%t nesXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;x<z<t<y||z<x<y<t];
%t Table[Length[Select[sps[Range[n]],nesXQ[#]&&croXQ[#]&]],{n,0,8}]
%Y Crossing and nesting set partitions are (both) A016098.
%Y Crossing, capturing set partitions are A326246.
%Y Nesting, non-crossing set partitions are A122880.
%Y Cf. A000108, A000110, A001519, A058681, A099947, A117662, A324170.
%Y Cf. A326209, A326211, A326243, A326245, A326256, A326258.
%K nonn,more
%O 0,6
%A _Gus Wiseman_, Jun 20 2019
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