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A324328 Number of topologically connected chord graphs on a subset of {1,...,n}. 11
1, 1, 2, 4, 8, 27, 354 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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.
LINKS
FORMULA
Binomial transform of A324327.
EXAMPLE
The a(0) = 1 through a(5) = 27 graphs:
{} {} {} {} {} {}
{{12}} {{12}} {{12}} {{12}}
{{13}} {{13}} {{13}}
{{23}} {{14}} {{14}}
{{23}} {{15}}
{{24}} {{23}}
{{34}} {{24}}
{{13}{24}} {{25}}
{{34}}
{{35}}
{{45}}
{{13}{24}}
{{13}{25}}
{{14}{25}}
{{14}{35}}
{{24}{35}}
{{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}]], Length[crosscmpts[#]]<=1&]], {n, 0, 5}]
CROSSREFS
Cf. A000108, A000699, A001764, A002061, A007297, A016098, A054726 (non-crossing chord graphs), A099947, A136653, A268814.
Cf. A324168, A324169, A324172, A324173, A324323, A324327 (covering case).
Sequence in context: A212409 A006399 A326293 * A056800 A302915 A259135
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 22 2019
STATUS
approved

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Last modified July 17 10:03 EDT 2024. Contains 374375 sequences. (Running on oeis4.)