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A324327
Number of topologically connected chord graphs covering {1,...,n}.
14
1, 0, 1, 0, 1, 11, 257
OFFSET
0,6
COMMENTS
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.
Covering means there are no isolated vertices.
FORMULA
Inverse binomial transform of A324328.
EXAMPLE
The a(0) = 1 through a(5) = 11 graphs:
{} {{12}} {{13}{24}} {{13}{14}{25}}
{{13}{24}{25}}
{{13}{24}{35}}
{{14}{24}{35}}
{{14}{25}{35}}
{{13}{14}{24}{25}}
{{13}{14}{24}{35}}
{{13}{14}{25}{35}}
{{13}{24}{25}{35}}
{{14}{24}{25}{35}}
{{13}{14}{24}{25}{35}}
MATHEMATICA
croXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
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]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn, {1}], Select[Subsets[stn, {2}], croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], And[Union@@#==Range[n], Length[crosscmpts[#]]<=1]&]], {n, 0, 5}]
CROSSREFS
Cf. A000108, A000699 (the case with disjoint edges), A001764, A002061, A007297, A016098, A054726, A099947, A136653 (the case with set-theoretical connectedness also), A268814.
Cf. A324167, A324169 (non-crossing covers), A324172, A324173, A324323, A324328 (non-covering case).
Sequence in context: A052782 A186249 A186250 * A142208 A168466 A091159
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 22 2019
STATUS
approved