%I #13 Feb 19 2019 00:09:00
%S 1,0,1,0,1,1,0,1,3,1,0,2,6,6,1,0,6,15,20,10,1,0,21,51,65,50,15,1,0,85,
%T 203,252,210,105,21,1,0,385,912,1120,938,560,196,28,1,0,1907,4527,
%U 5520,4620,2898,1302,336,36,1,0,10205,24370,29700,24780,15792,7812,2730,540,45,1
%N Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.
%C A set partition is crossing if it contains a pair of blocks of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.
%C The topologically connected components of a set partition correspond to the blocks of its minimal non-crossing coarsening.
%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000925">The number of topologically connected components of a set partition</a>.
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 3 1
%e 0 2 6 6 1
%e 0 6 15 20 10 1
%e 0 21 51 65 50 15 1
%e 0 85 203 252 210 105 21 1
%e 0 385 912 1120 938 560 196 28 1
%e 0 1907 4527 5520 4620 2898 1302 336 36 1
%e 0 10205 24370 29700 24780 15792 7812 2730 540 45 1
%e Row n = 4 counts the following set partitions:
%e {{1234}} {{1}{234}} {{1}{2}{34}} {{1}{2}{3}{4}}
%e {{13}{24}} {{12}{34}} {{1}{23}{4}}
%e {{123}{4}} {{12}{3}{4}}
%e {{124}{3}} {{1}{24}{3}}
%e {{134}{2}} {{13}{2}{4}}
%e {{14}{23}} {{14}{2}{3}}
%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[#]]==k&]],{n,0,8},{k,0,n}]
%Y Row sums are A000110. Column k = 1 is A099947.
%Y Cf. A000108, A001263, A002061, A002662, A007297, A016098, A048993, A054726, A293510, A305078, A305079, A323818.
%Y Cf. A324166, A324167, A324169, A324170, A324171, A324172.
%K nonn,tabl
%O 0,9
%A _Gus Wiseman_, Feb 17 2019