cp's OEIS Frontend

A324328 Number of topologically connected chord graphs on a subset of {1,...,n}.

%I A324328 #6 Feb 22 2019 21:17:02
%S A324328 1,1,2,4,8,27,354
%N A324328 Number of topologically connected chord graphs on a subset of {1,...,n}.
%C A324328 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.
%F A324328 Binomial transform of A324327.
%e A324328 The a(0) = 1 through a(5) = 27 graphs:
%e A324328   {}  {}  {}      {}      {}          {}
%e A324328           {{12}}  {{12}}  {{12}}      {{12}}
%e A324328                   {{13}}  {{13}}      {{13}}
%e A324328                   {{23}}  {{14}}      {{14}}
%e A324328                           {{23}}      {{15}}
%e A324328                           {{24}}      {{23}}
%e A324328                           {{34}}      {{24}}
%e A324328                           {{13}{24}}  {{25}}
%e A324328                                       {{34}}
%e A324328                                       {{35}}
%e A324328                                       {{45}}
%e A324328                                       {{13}{24}}
%e A324328                                       {{13}{25}}
%e A324328                                       {{14}{25}}
%e A324328                                       {{14}{35}}
%e A324328                                       {{24}{35}}
%e A324328                                       {{13}{14}{25}}
%e A324328                                       {{13}{24}{25}}
%e A324328                                       {{13}{24}{35}}
%e A324328                                       {{14}{24}{35}}
%e A324328                                       {{14}{25}{35}}
%e A324328                                       {{13}{14}{24}{25}}
%e A324328                                       {{13}{14}{24}{35}}
%e A324328                                       {{13}{14}{25}{35}}
%e A324328                                       {{13}{24}{25}{35}}
%e A324328                                       {{14}{24}{25}{35}}
%e A324328                                       {{13}{14}{24}{25}{35}}
%t A324328 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324328 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 A324328 crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
%t A324328 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[crosscmpts[#]]<=1&]],{n,0,5}]
%Y A324328 Cf. A000108, A000699, A001764, A002061, A007297, A016098, A054726 (non-crossing chord graphs), A099947, A136653, A268814.
%Y A324328 Cf. A324168, A324169, A324172, A324173, A324323, A324327 (covering case).
%K A324328 nonn,more
%O A324328 0,3
%A A324328 _Gus Wiseman_, Feb 22 2019