%I #16 Jul 04 2019 13:57:25

%S 1,0,1,4,13,44,149,504,1705,5768,19513,66012

%N Number of simple graphs covering {1..n} with no crossing or nesting edges.

%C 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, and nesting if a < c < d < b or c < a < b < d.

%C Is this (apart from offsets) the same as A073717? - _R. J. Mathar_, Jul 04 2019

%H Eric Marberg, <a href="http://arxiv.org/abs/1203.5738">Crossings and nestings in colored set partitions</a>, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

%H Gus Wiseman, <a href="/A326329/a326329.png">The a(5) = 44 covering simple graphs with no crossing or nesting edges</a>.

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&!MatchQ[#,{___,{x_,y_},___,{z_,t_},___}/;x<z<y<t||z<x<t<y||x<z<t<y||z<x<y<t]&]],{n,0,5}]

%Y The case for set partitions is A001519.

%Y Covering simple graphs are A006129.

%Y The case with just nesting or just crossing edges forbidden is A324169.

%Y The binomial transform is the non-covering case A326244.

%Y Cf. A000108, A006125, A007297, A054726, A099947, A324327, A326279, A326293.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Jun 27 2019