A326341 Number of minimal topologically connected chord graphs covering {1..n}.

1, 0, 1, 0, 1, 5, 22, 119

Offset

0,6

Comments

Covering means there are no isolated vertices. Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b. A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.

Links

Table of n, a(n) for n=0..7.

Example

The a(4) = 1 through a(6) = 22 edge-sets:
  {13,24}  {13,14,25}  {13,25,46}
           {13,24,25}  {14,25,36}
           {13,24,35}  {14,26,35}
           {14,24,35}  {15,24,36}
           {14,25,35}  {13,14,15,26}
                       {13,14,25,26}
                       {13,15,24,26}
                       {13,15,26,46}
                       {13,24,25,26}
                       {13,24,25,36}
                       {13,24,26,35}
                       {13,24,35,36}
                       {13,24,35,46}
                       {14,15,26,36}
                       {14,24,35,36}
                       {14,24,35,46}
                       {14,25,35,46}
                       {15,24,35,46}
                       {15,25,35,46}
                       {15,25,36,46}
                       {15,26,35,46}
                       {15,26,36,46}

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[fasmin[Select[Subsets[Subsets[Range[n], {2}]], And[Union@@#==Range[n], Length[crosscmpts[#]]<=1]&]]], {n, 0, 5}]

Crossrefs

The non-minimal case is A324327.

Minimal covers are A053530.

Topologically connected graphs are A324327 (covering) or A324328 (all).

Cf. A000108, A006125, A007297, A054726, A136653, A324169, A326210, A326293.

Sequence in context: A020003 A276750 A131460 * A062794 A036235 A159596

Adjacent sequences: A326338 A326339 A326340 * A326342 A326343 A326344

Keyword

nonn,more

Author

Gus Wiseman, Jun 29 2019

Status

approved