A324327 Number of topologically connected chord graphs covering {1,...,n}.

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.

Links

Table of n, a(n) for n=0..6.

Gus Wiseman, The a(5) = 11 topologically connected chord graphs.

Gus Wiseman, The a(6) = 257 topologically connected chord graphs.

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

Adjacent sequences: A324324 A324325 A324326 * A324328 A324329 A324330

Keyword

nonn,more

Author

Gus Wiseman, Feb 22 2019

Status

approved