A324166 - OEIS

%I #7 Feb 19 2019 00:07:30

%S 1,1,1,1,2,6,18,57,207,842,3673,17062,84897

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

%C 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 The a(6) = 18 totally crossing set partitions:

%e   {{1,2,3,4,5,6}}

%e   {{1,4,6},{2,3,5}}

%e   {{1,4,5},{2,3,6}}

%e   {{1,3,6},{2,4,5}}

%e   {{1,3,5},{2,4,6}}

%e   {{1,3,4},{2,5,6}}

%e   {{1,2,5},{3,4,6}}

%e   {{1,2,4},{3,5,6}}

%e   {{4,6},{1,2,3,5}}

%e   {{3,6},{1,2,4,5}}

%e   {{3,5},{1,2,4,6}}

%e   {{2,6},{1,3,4,5}}

%e   {{2,5},{1,3,4,6}}

%e   {{2,4},{1,3,5,6}}

%e   {{1,5},{2,3,4,6}}

%e   {{1,4},{2,3,5,6}}

%e   {{1,3},{2,4,5,6}}

%e   {{1,4},{2,5},{3,6}}

%t nn=6;

%t nonXQ[stn_]:=!MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];

%t 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 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]!={}||nonXQ[{#1,#2}]&],Union@@#==Range[n]&]],{n,0,nn}]

%Y Cf. A000108 (non-crossing partitions), A000110, A000296, A002662, A016098 (crossing partitions), A054726, A099947 (topologically connected partitions), A305854, A306006, A306418, A306438, A319752.

%Y Cf. A324170, A324171, A324172, A324173.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Feb 17 2019