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%I #7 Feb 22 2019 21:16:21
%S 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 0,0,0,0,1,99,258,27,0,0,0,0,0,1,219,1222,465,0,0,0,0,0
%N 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 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 Triangle begins:
%e 1
%e 0 1
%e 0 1 0
%e 0 1 0 0
%e 0 1 1 0 0
%e 0 1 5 0 0 0
%e 0 1 16 4 0 0 0
%e 0 1 42 42 0 0 0 0
%e 0 1 99 258 27 0 0 0 0
%e 0 1 219 1222 465 0 0 0 0 0
%e Row n = 6 counts the following set partitions:
%e {{123456}} {{1235}{46}} {{13}{25}{46}}
%e {{124}{356}} {{14}{25}{36}}
%e {{1245}{36}} {{14}{26}{35}}
%e {{1246}{35}} {{15}{24}{36}}
%e {{125}{346}}
%e {{13}{2456}}
%e {{134}{256}}
%e {{1345}{26}}
%e {{1346}{25}}
%e {{135}{246}}
%e {{1356}{24}}
%e {{136}{245}}
%e {{14}{2356}}
%e {{145}{236}}
%e {{146}{235}}
%e {{15}{2346}}
%t croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t 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 crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t Table[Length[Select[sps[Range[n]],Length[crosscmpts[#]]<=1&&Length[#]==k&]],{n,0,6},{k,0,n}]
%Y Row sums are A099947. Row k = 2 is A002662.
%Y Cf. A000108, A000110, A001263, A016098, A136653, A268814, A268815, A306438, A324011.
%Y Cf. A324166, A324172, A324173, A324327, A324328.
%K nonn,tabl,more
%O 0,18
%A _Gus Wiseman_, Feb 22 2019
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