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COMMENTS
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Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d. A graph has its nesting edges connected if the graph whose vertices are the edges and whose edges are nesting pairs of edges is connected.
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MATHEMATICA
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nesXQ[stn_]:=MatchQ[stn, {___, {x_, y_}, ___, {z_, t_}, ___}/; x<z<t<y||z<x<y<t];
nestcmpts[stn_]:=csm[Union[List/@stn, Select[Subsets[stn, {2}], nesXQ]]];
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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[nestcmpts[#]]<=1&]], {n, 0, 5}]
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CROSSREFS
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The covering case is the inverse binomial transform A326331.
Graphs whose crossing edges are connected are A324328.
Cf. A006125, A007297, A054726, A099947, A117662, A136653, A324328.
Cf. A326210, A326293, A326335, A326336, A326337, A326338, A326339.
Sequence in context: A018346 A006398 A053364 * A018355 A100083 A151406
Adjacent sequences: A326327 A326328 A326329 * A326331 A326332 A326333
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