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A326248 Number of crossing, nesting set partitions of {1..n}.

%I A326248 #10 Oct 16 2024 20:16:33
%S A326248 0,0,0,0,0,2,28,252,1890,13020,86564,571944,3826230,26233662,
%T A326248 185746860,1364083084,10410773076,82609104802,681130756224,
%U A326248 5829231836494,51711093240518,474821049202852,4506533206814480,44151320870760216,445956292457725714
%N A326248 Number of crossing, nesting set partitions of {1..n}.
%C A326248 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.
%H A326248 Christian Sievers, <a href="/A326248/b326248.txt">Table of n, a(n) for n = 0..576</a>
%F A326248 a(n) = A000110(n) - 2*A000108(n) + A001519(n). - _Christian Sievers_, Oct 16 2024
%e A326248 The a(5) = 2 set partitions:
%e A326248   {{1,4},{2,3,5}}
%e A326248   {{1,3,4},{2,5}}
%t A326248 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A326248 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A326248 nesXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;x<z<t<y||z<x<y<t];
%t A326248 Table[Length[Select[sps[Range[n]],nesXQ[#]&&croXQ[#]&]],{n,0,8}]
%Y A326248 Crossing and nesting set partitions are (both) A016098.
%Y A326248 Crossing, capturing set partitions are A326246.
%Y A326248 Nesting, non-crossing set partitions are A122880.
%Y A326248 Cf. A000108, A000110, A001519, A058681, A099947, A117662, A324170.
%Y A326248 Cf. A326209, A326211, A326243, A326245, A326256, A326258.
%K A326248 nonn
%O A326248 0,6
%A A326248 _Gus Wiseman_, Jun 20 2019
%E A326248 a(11) and beyond from _Christian Sievers_, Oct 16 2024