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A054726 Number of graphs with n nodes on a circle without crossing edges. 55
1, 1, 2, 8, 48, 352, 2880, 25216, 231168, 2190848, 21292032, 211044352, 2125246464, 21681954816, 223623069696, 2327818174464, 24424842461184, 258054752698368, 2742964283768832, 29312424612462592, 314739971287154688, 3393951437605044224, 36739207546043105280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

From Gus Wiseman, Jun 22 2019: (Start)

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}

Cf. A001519, A117662, A326244, A326256, A326257, A326279.

(End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.

F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).

Michael Drmota, Anna de Mier, Marc Noy, Extremal statistics on non-crossing configurations, Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (3). - N. J. A. Sloane, May 18 2014

P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discr. Math. 204 (1999), 203-229

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486

Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 10.

Marco Kuhlmann, Tabulation of Noncrossing Acyclic Digraphs, arXiv preprint arXiv:1504.04993 [cs.DS], 2015.

Gus Wiseman, The a(4) = 48 non-crossing graphs.

Gus Wiseman, The a(5) = 352 non-crossing graphs.

Gus Wiseman, The a(4) = 48 non-nesting simple graphs.

Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

Sequences related to chord diagrams

FORMULA

a(n) = 2^n*A001003(n-2) for n>2.

From Lee A. Newberg, Aug 09 2011: (Start)

G.f.: 1 + (3/2)*z - z^2 - (z/2)*sqrt(1 - 12*z + 4*z^2);

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

MAPLE

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)];

MATHEMATICA

Join[{a = 1, b = 1}, Table[c = (6*(2*n - 3)*b)/n - (4*(n - 3) a)/n; a = b; b = c, {n, 1, 40}]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)

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 *)

PROG

(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

CROSSREFS

Cf. A006013, A001003.

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

Cf. A000108 (non-crossing set partitions), A000124, A006125, A007297 (connected case), A194560, A306438, A324167, A324169 (covering case), A324173, A326210.

Sequence in context: A171455 A136722 A085615 * A003576 A225042 A326887

Adjacent sequences:  A054723 A054724 A054725 * A054727 A054728 A054729

KEYWORD

nonn

AUTHOR

Philippe Flajolet, Apr 20 2000

EXTENSIONS

Offset changed to 0 by Lee A. Newberg, Aug 03 2011

STATUS

approved

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Last modified December 11 23:44 EST 2019. Contains 329945 sequences. (Running on oeis4.)