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%I #6 Feb 22 2019 21:16:53
%S 1,0,1,0,1,11,257
%N Number of topologically connected chord graphs covering {1,...,n}.
%C A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected, where two edges cross each other if they are of the form {{x,y},{z,t}} with x < z < y < t or z < x < t < y.
%C Covering means there are no isolated vertices.
%H Gus Wiseman, <a href="/A324327/a324327.png">The a(5) = 11 topologically connected chord graphs.</a>
%H Gus Wiseman, <a href="/A324327/a324327_1.png">The a(6) = 257 topologically connected chord graphs.</a>
%F Inverse binomial transform of A324328.
%e The a(0) = 1 through a(5) = 11 graphs:
%e {} {{12}} {{13}{24}} {{13}{14}{25}}
%e {{13}{24}{25}}
%e {{13}{24}{35}}
%e {{14}{24}{35}}
%e {{14}{25}{35}}
%e {{13}{14}{24}{25}}
%e {{13}{14}{24}{35}}
%e {{13}{14}{25}{35}}
%e {{13}{24}{25}{35}}
%e {{14}{24}{25}{35}}
%e {{13}{14}{24}{25}{35}}
%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 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[crosscmpts[#]]<=1]&]],{n,0,5}]
%Y Cf. A000108, A000699 (the case with disjoint edges), A001764, A002061, A007297, A016098, A054726, A099947, A136653 (the case with set-theoretical connectedness also), A268814.
%Y Cf. A324167, A324169 (non-crossing covers), A324172, A324173, A324323, A324328 (non-covering case).
%K nonn,more
%O 0,6
%A _Gus Wiseman_, Feb 22 2019