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A324169 Number of labeled graphs covering the vertex set {1,...,n} with no crossing edges.

%I A324169 #9 Jan 20 2023 16:16:47
%S A324169 1,0,1,4,25,176,1353,11012,93329,815104,7285489,66324644,612863337,
%T A324169 5733381616,54195878137,516852285668,4966883732129,48049936644736,
%U A324169 467566946973537,4573486005681092,44942852084894777,443484037981300144,4392621673072766505
%N A324169 Number of labeled graphs covering the vertex set {1,...,n} with no crossing edges.
%C A324169 Two edges {x,y}, {z,t} are crossing if either x < z < y < t or z < x < t < y. If the vertices are arranged in a circle, this is equivalent to crossing of chords.
%C A324169 Covering means there are no isolated vertices.
%H A324169 Andrew Howroyd, <a href="/A324169/b324169.txt">Table of n, a(n) for n = 0..500</a>
%H A324169 Gus Wiseman, <a href="/A324169/a324169.png">The a(4) = 25 graphs covering {1,2,3,4} with no crossing edges.</a>
%H A324169 Gus Wiseman, <a href="/A324169/a324169_1.png">The a(5) = 176 graphs covering 5 vertices with no crossing edges, radial embedding.</a>
%F A324169 Inverse binomial transform of A054726.
%F A324169 G.f.: (2 + 7*x + 3*x^2 - x*sqrt(1 - 10*x - 7*x^2))/(2*(1 + x)^3). - _Andrew Howroyd_, Jan 20 2023
%t A324169 nn=8;
%t A324169 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324169 stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
%t A324169 Table[Length[Select[stableSets[Subsets[Range[n],{2}],croXQ[{#1,#2}]&],Union@@#==Range[n]&]],{n,0,nn}]
%o A324169 (PARI) seq(n)=Vec((2 + 7*x + 3*x^2 - x*sqrt(1 - 10*x - 7*x^2 + O(x^n)))/(2*(1 + x)^3)) \\ _Andrew Howroyd_, Jan 20 2023
%Y A324169 Cf. A000108, A000124, A001006, A001764, A003465, A007297 (connected case), A016098, A054726 (non-crossing graphs), A099947, A306438.
%Y A324169 Cf. A324167, A324171, A324173.
%K A324169 nonn
%O A324169 0,4
%A A324169 _Gus Wiseman_, Feb 17 2019