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A054726
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Number of graphs with n nodes on a circle without crossing edges.
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56
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1, 1, 2, 8, 48, 352, 2880, 25216, 231168, 2190848, 21292032, 211044352, 2125246464, 21681954816, 223623069696, 2327818174464, 24424842461184, 258054752698368, 2742964283768832, 29312424612462592, 314739971287154688, 3393951437605044224, 36739207546043105280
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OFFSET
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0,3
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COMMENTS
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Related to Schröder's second problem.
A001006 gives number of ways of drawing any number of nonintersecting chords between n points on a circle, while this sequence gives number of ways of drawing noncrossing chords between n points on a circle. The difference is that nonintersection chords have no point in common, while noncrossing chords may share an endpoint. - David W. Wilson, Jan 30 2003
For n>0, a(n) = number of lattice paths from (0,0) to (n-1,n-1) that consist of steps (i,j), i,j nonnegative integers not both 0 and that stay strictly below the line y=x except at their endpoints. For example, a(3)=8 counts the paths with following step sequences: {(2, 2)}, {(2, 1), (0, 1)}, {(2, 0), (0, 2)}, {(2, 0), (0, 1), (0, 1)}, {(1, 0), (1, 2)}, {(1, 0), (1, 1), (0, 1)}, {(1, 0), (1, 0), (0, 2)}, {(1, 0), (1, 0), (0, 1), (0, 1)}. If the word "strictly" is replaced by "weakly", the counting sequence becomes A059435. - David Callan, Jun 07 2006
The nodes on the circle are distinguished by their positions but are otherwise unlabeled. - Lee A. Newberg, Aug 09 2011
Conjecture: Also the number of simple graphs with vertices {1..n} not containing any pair of nesting edges. Two edges {a,b}, {c,d} where a < b and c < d are nesting if a < c and b > d or a > c and b < d. For example, the a(0) = 1 through a(3) = 8 non-nesting edge-sets are:
{} {} {} {}
{12} {12}
{13}
{23}
{12,13}
{12,23}
{13,23}
{12,13,23}
(End)
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LINKS
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FORMULA
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G.f.: 1 + (3/2)*z - z^2 - (z/2)*sqrt(1 - 12*z + 4*z^2);
D-finite with recurrence: a(n) = ((12*n-30)*a(n-1) - (4*n-16)*a(n-2)) / (n-1) for n>1. (End)
a(n) ~ 2^(n - 7/4) * (1 + sqrt(2))^(2*n-3) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 11 2012, simplified Dec 24 2017
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MAPLE
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with(combstruct): br:= {EA = Union(Sequence(EA, card >= 2), Prod(V, Sequence(EA), Sequence(EA))), V=Union(Prod(Z, G)), G=Union(Epsilon, Prod(Z, G), Prod(V, V, Sequence(EA), Sequence(EA), Sequence(Union(Sequence(EA, card>=1), Prod(V, Sequence(EA), Sequence(EA)))))) }; ggSeq := [seq(count([G, br], size=i), i=0..20)];
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MATHEMATICA
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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[stableSets[Subsets[Range[n], {2}], croXQ[{#1, #2}]&]], {n, 0, nn}] (* Gus Wiseman, Feb 19 2019 *)
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PROG
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(PARI) z='z+O('z^66); Vec( 1+3/2*z-z^2-z/2*sqrt(1-12*z+4*z^2) ) \\ Joerg Arndt, Mar 01 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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