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A326293 Number of non-nesting, topologically connected simple graphs with vertices {1..n}. 16
1, 1, 2, 4, 8, 27, 192, 1750 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d. A graph with positive integer vertices is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.
LINKS
MATHEMATICA
croXQ[eds_]:=MatchQ[eds, {___, {x_, y_}, ___, {z_, t_}, ___}/; x<z<y<t||z<x<t<y];
nesXQ[eds_]:=MatchQ[eds, {___, {x_, y_}, ___, {z_, t_}, ___}/; x<z<t<y||z<x<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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], !nesXQ[#]&&Length[csm[Union[Subsets[#, {1}], Select[Subsets[#, {2}], croXQ]]]]<=1&]], {n, 0, 5}]
CROSSREFS
The inverse binomial transform is the covering case A326349.
Topologically connected simple graphs are A324328.
Non-crossing simple graphs are A054726.
Topologically connected set partitions are A099947.
Sequence in context: A037170 A212409 A006399 * A324328 A056800 A302915
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 29 2019
STATUS
approved

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Last modified July 24 02:09 EDT 2024. Contains 374575 sequences. (Running on oeis4.)