A324169 - OEIS

A324169

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

1, 0, 1, 4, 25, 176, 1353, 11012, 93329, 815104, 7285489, 66324644, 612863337, 5733381616, 54195878137, 516852285668, 4966883732129, 48049936644736, 467566946973537, 4573486005681092, 44942852084894777, 443484037981300144, 4392621673072766505

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OFFSET

0,4

COMMENTS

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.

Covering means there are no isolated vertices.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..500

Gus Wiseman, The a(4) = 25 graphs covering {1,2,3,4} with no crossing edges.

Gus Wiseman, The a(5) = 176 graphs covering 5 vertices with no crossing edges, radial embedding.

FORMULA

Inverse binomial transform of A054726.

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

MATHEMATICA

nn=8;

croXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];

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]]]];

Table[Length[Select[stableSets[Subsets[Range[n], {2}], croXQ[{#1, #2}]&], Union@@#==Range[n]&]], {n, 0, nn}]

PROG

(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

CROSSREFS

Cf. A000108, A000124, A001006, A001764, A003465, A007297 (connected case), A016098, A054726 (non-crossing graphs), A099947, A306438.

Cf. A324167, A324171, A324173.

Sequence in context: A006348 A213608 A369325 * A213231 A051820 A362206

Adjacent sequences: A324166 A324167 A324168 * A324170 A324171 A324172

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 17 2019

STATUS

approved