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A324173 Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.

%I A324173 #13 Feb 19 2019 00:09:00
%S A324173 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 A324173 203,252,210,105,21,1,0,385,912,1120,938,560,196,28,1,0,1907,4527,
%U A324173 5520,4620,2898,1302,336,36,1,0,10205,24370,29700,24780,15792,7812,2730,540,45,1
%N A324173 Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.
%C A324173 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 A324173 The topologically connected components of a set partition correspond to the blocks of its minimal non-crossing coarsening.
%H A324173 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000925">The number of topologically connected components of a set partition</a>.
%e A324173 Triangle begins:
%e A324173      1
%e A324173      0     1
%e A324173      0     1     1
%e A324173      0     1     3     1
%e A324173      0     2     6     6     1
%e A324173      0     6    15    20    10     1
%e A324173      0    21    51    65    50    15     1
%e A324173      0    85   203   252   210   105    21     1
%e A324173      0   385   912  1120   938   560   196    28     1
%e A324173      0  1907  4527  5520  4620  2898  1302   336    36     1
%e A324173      0 10205 24370 29700 24780 15792  7812  2730   540    45     1
%e A324173 Row n = 4 counts the following set partitions:
%e A324173   {{1234}}    {{1}{234}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
%e A324173   {{13}{24}}  {{12}{34}}  {{1}{23}{4}}
%e A324173               {{123}{4}}  {{12}{3}{4}}
%e A324173               {{124}{3}}  {{1}{24}{3}}
%e A324173               {{134}{2}}  {{13}{2}{4}}
%e A324173               {{14}{23}}  {{14}{2}{3}}
%t A324173 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324173 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 A324173 crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
%t A324173 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A324173 Table[Length[Select[sps[Range[n]],Length[crosscmpts[#]]==k&]],{n,0,8},{k,0,n}]
%Y A324173 Row sums are A000110. Column k = 1 is A099947.
%Y A324173 Cf. A000108, A001263, A002061, A002662, A007297, A016098, A048993, A054726, A293510, A305078, A305079, A323818.
%Y A324173 Cf. A324166, A324167, A324169, A324170, A324171, A324172.
%K A324173 nonn,tabl
%O A324173 0,9
%A A324173 _Gus Wiseman_, Feb 17 2019