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A324323 Regular triangle read by rows where T(n,k) is the number of topologically connected set partitions of {1,...,n} with k blocks, 0 <= k <= n.

%I A324323 #7 Feb 22 2019 21:16:21
%S A324323 1,0,1,0,1,0,0,1,0,0,0,1,1,0,0,0,1,5,0,0,0,0,1,16,4,0,0,0,0,1,42,42,0,
%T A324323 0,0,0,0,1,99,258,27,0,0,0,0,0,1,219,1222,465,0,0,0,0,0
%N A324323 Regular triangle read by rows where T(n,k) is the number of topologically connected set partitions of {1,...,n} with k blocks, 0 <= k <= n.
%C A324323 A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y.
%e A324323 Triangle begins:
%e A324323     1
%e A324323     0    1
%e A324323     0    1    0
%e A324323     0    1    0    0
%e A324323     0    1    1    0    0
%e A324323     0    1    5    0    0    0
%e A324323     0    1   16    4    0    0    0
%e A324323     0    1   42   42    0    0    0    0
%e A324323     0    1   99  258   27    0    0    0    0
%e A324323     0    1  219 1222  465    0    0    0    0    0
%e A324323 Row n = 6 counts the following set partitions:
%e A324323   {{123456}}  {{1235}{46}}  {{13}{25}{46}}
%e A324323               {{124}{356}}  {{14}{25}{36}}
%e A324323               {{1245}{36}}  {{14}{26}{35}}
%e A324323               {{1246}{35}}  {{15}{24}{36}}
%e A324323               {{125}{346}}
%e A324323               {{13}{2456}}
%e A324323               {{134}{256}}
%e A324323               {{1345}{26}}
%e A324323               {{1346}{25}}
%e A324323               {{135}{246}}
%e A324323               {{1356}{24}}
%e A324323               {{136}{245}}
%e A324323               {{14}{2356}}
%e A324323               {{145}{236}}
%e A324323               {{146}{235}}
%e A324323               {{15}{2346}}
%t A324323 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324323 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A324323 crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
%t A324323 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A324323 Table[Length[Select[sps[Range[n]],Length[crosscmpts[#]]<=1&&Length[#]==k&]],{n,0,6},{k,0,n}]
%Y A324323 Row sums are A099947. Row k = 2 is A002662.
%Y A324323 Cf. A000108, A000110, A001263, A016098, A136653, A268814, A268815, A306438, A324011.
%Y A324323 Cf. A324166, A324172, A324173, A324327, A324328.
%K A324323 nonn,tabl,more
%O A324323 0,18
%A A324323 _Gus Wiseman_, Feb 22 2019