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A324166 Number of totally crossing set partitions of {1,...,n}.

%I A324166 #7 Feb 19 2019 00:07:30
%S A324166 1,1,1,1,2,6,18,57,207,842,3673,17062,84897
%N A324166 Number of totally crossing set partitions of {1,...,n}.
%C A324166 A set partition is totally crossing if every pair of distinct blocks is of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y.
%e A324166 The a(6) = 18 totally crossing set partitions:
%e A324166   {{1,2,3,4,5,6}}
%e A324166   {{1,4,6},{2,3,5}}
%e A324166   {{1,4,5},{2,3,6}}
%e A324166   {{1,3,6},{2,4,5}}
%e A324166   {{1,3,5},{2,4,6}}
%e A324166   {{1,3,4},{2,5,6}}
%e A324166   {{1,2,5},{3,4,6}}
%e A324166   {{1,2,4},{3,5,6}}
%e A324166   {{4,6},{1,2,3,5}}
%e A324166   {{3,6},{1,2,4,5}}
%e A324166   {{3,5},{1,2,4,6}}
%e A324166   {{2,6},{1,3,4,5}}
%e A324166   {{2,5},{1,3,4,6}}
%e A324166   {{2,4},{1,3,5,6}}
%e A324166   {{1,5},{2,3,4,6}}
%e A324166   {{1,4},{2,3,5,6}}
%e A324166   {{1,3},{2,4,5,6}}
%e A324166   {{1,4},{2,5},{3,6}}
%t A324166 nn=6;
%t A324166 nonXQ[stn_]:=!MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324166 stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
%t A324166 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]!={}||nonXQ[{#1,#2}]&],Union@@#==Range[n]&]],{n,0,nn}]
%Y A324166 Cf. A000108 (non-crossing partitions), A000110, A000296, A002662, A016098 (crossing partitions), A054726, A099947 (topologically connected partitions), A305854, A306006, A306418, A306438, A319752.
%Y A324166 Cf. A324170, A324171, A324172, A324173.
%K A324166 nonn,more
%O A324166 0,5
%A A324166 _Gus Wiseman_, Feb 17 2019