A054726 -id:A054726 - cp's OEIS frontend

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

Offset: 0

Gus Wiseman, Feb 17 2019

nonn

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.

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.

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

Cf. A324167, A324171, A324173.

nn=8;
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

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

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

A006533 Place n equally-spaced points around a circle and join every pair of points by a chord; this divides the circle into a(n) regions.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 57, 88, 163, 230, 386, 456, 794, 966, 1471, 1712, 2517, 2484, 4048, 4520, 6196, 6842, 9109, 9048, 12951, 14014, 17902, 19208, 24158, 21510, 31931, 33888, 41449, 43826, 52956, 52992, 66712, 70034, 82993, 86840, 102091, 97776, 124314, 129448, 149986, 155894, 179447, 179280

Offset: 1

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

This sequence and A007678 are two equivalent ways of presenting the same sequence. - N. J. A. Sloane, Jan 23 2020

In contrast to A007678, which only counts the polygons, this sequence also counts the n segments of the circle bounded by the arc of the circle and the straight line, both joining two neighboring points on the circle. Therefore a(n) = A007678(n) + n. - M. F. Hasler, Dec 12 2021

Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. F. Hasler, Interactive illustration of A006561(n) & A006533(n); colored version for n=6 and for n=8.
Sascha Kurz, m-gons in regular n-gons
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Burkard Polster (Mathologer), The hardest "What comes next?" (Euler's pentagonal formula), Youtube video, Oct 17 2020.
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, No. 1 (1998), pp. 135-156. (Author's copy.).
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
Bjorn Poonen and Michael Rubinstein, Mathematica programs for these sequences
Michael Rubinstein, Drawings for n=4,5,6,...
Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
Sequences formed by drawing all diagonals in regular polygon

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

del[m_,n_]:=If[Mod[n,m]==0,1,0];
R[n_]:=(n^4-6n^3+23n^2-18n+24)/24 + del[2,n](-5n^3+42n^2-40n-48)/48 - del[4,n](3n/4) + del[6,n](-53n^2+310n)/12 + del[12,n](49n/2) + del[18,n]*32n + del[24,n]*19n - del[30,n]*36n - del[42,n]*50n - del[60,n]*190n - del[84,n]*78n - del[90,n]*48n - del[120,n]*78n - del[210,n]*48n;
Table[R[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)

apply( {A006533(n)=if(n%2, (((n-6)*n+23)*n-18)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 19, 28) +!(n%6)*(310 -53*n))/12 +!(n%12)*49/2 +!(n%18)*32 +!(n%24)*19 -!(n%30)*36 -!(n%42)*50 -!(n%60)*190 -!(n%84)*78 -!(n%90)*48 -!(n%120)*78 -!(n%210)*48)*n)}, [1..44]) \\ M. F. Hasler, Aug 06 2021

def d(n,m): return not n % m
def A006533(n): return (1176*d(n,12)*n - 3744*d(n,120)*n + 1536*d(n,18)*n - d(n,2)*(5*n**3 - 42*n**2 + 40*n + 48) - 2304*d(n,210)*n + 912*d(n,24)*n - 1728*d(n,30)*n - 36*d(n,4)*n - 2400*d(n,42)*n - 4*d(n,6)*n*(53*n - 310) - 9120*d(n,60)*n - 3744*d(n,84)*n - 2304*d(n,90)*n + 2*n**4 - 12*n**3 + 46*n**2 - 36*n)//48 + 1 # Chai Wah Wu, Mar 08 2021

Poonen and Rubinstein give an explicit formula for a(n) (see Mma code).

a(n) = A007678(n) + n. - T. D. Noe, Dec 23 2006

Added more terms from b-file. - N. J. A. Sloane, Jan 23 2020

Edited definition. - N. J. A. Sloane, Mar 17 2024

A324173 Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 2, 6, 6, 1, 0, 6, 15, 20, 10, 1, 0, 21, 51, 65, 50, 15, 1, 0, 85, 203, 252, 210, 105, 21, 1, 0, 385, 912, 1120, 938, 560, 196, 28, 1, 0, 1907, 4527, 5520, 4620, 2898, 1302, 336, 36, 1, 0, 10205, 24370, 29700, 24780, 15792, 7812, 2730, 540, 45, 1

Offset: 0

Gus Wiseman, Feb 17 2019

A set partition is crossing if it contains a pair of blocks of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

The topologically connected components of a set partition correspond to the blocks of its minimal non-crossing coarsening.

			Triangle begins:
     1
     0     1
     0     1     1
     0     1     3     1
     0     2     6     6     1
     0     6    15    20    10     1
     0    21    51    65    50    15     1
     0    85   203   252   210   105    21     1
     0   385   912  1120   938   560   196    28     1
     0  1907  4527  5520  4620  2898  1302   336    36     1
     0 10205 24370 29700 24780 15792  7812  2730   540    45     1
Row n = 4 counts the following set partitions:
  {{1234}}    {{1}{234}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
  {{13}{24}}  {{12}{34}}  {{1}{23}{4}}
              {{123}{4}}  {{12}{3}{4}}
              {{124}{3}}  {{1}{24}{3}}
              {{134}{2}}  {{13}{2}{4}}
              {{14}{23}}  {{14}{2}{3}}

FindStat, The number of topologically connected components of a set partition.

Row sums are A000110. Column k = 1 is A099947.
Cf. A000108, A001263, A002061, A002662, A007297, A016098, A048993, A054726, A293510, A305078, A305079, A323818.
Cf. A324166, A324167, A324169, A324170, A324171, A324172.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Length[crosscmpts[#]]==k&]],{n,0,8},{k,0,n}]

A006600 Total number of triangles visible in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956, 11297, 17234, 25935, 37424, 53516, 73404, 101745, 136200, 181279, 236258, 306383, 389264, 495650, 620048, 772785, 951384, 1167453, 1410350, 1716191, 2058848, 2463384, 2924000, 3462305, 4067028, 4776219, 5568786, 6479551

Offset: 3

N. J. A. Sloane

Place n equally-spaced points on a circle, join them in all possible ways; how many triangles can be seen?

			a(4) = 8 because in a quadrilateral the diagonals cross to make four triangles, which pair up to make four more.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=3..1000
Sascha Kurz, m-gons in regular n-gons
Victor Meally, Letter to N. J. A. Sloane, no date.
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv version, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
M. Rubinstein, Drawings for n=4,5,6,...
T. Sillke, Number of triangles for convex n-gon
S. E. Sommars and T. Sommars, Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, J. Integer Sequences, 1 (1998), #98.1.5.
Sequences formed by drawing all diagonals in regular polygon

Often confused with A005732.
Cf. A203016, A260417.
Row sums of A363174.
Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

del[m_,n_]:=If[Mod[n,m]==0,1,0]; Tri[n_]:=n(n-1)(n-2)(n^3+18n^2-43n+60)/720 - del[2,n](n-2)(n-7)n/8 - del[4,n](3n/4) - del[6,n](18n-106)n/3 + del[12,n]*33n + del[18,n]*36n + del[24,n]*24n - del[30,n]*96n - del[42,n]*72n - del[60,n]*264n - del[84,n]*96n - del[90,n]*48n - del[120,n]*96n - del[210,n]*48n; Table[Tri[n], {n,3,1000}] (* T. D. Noe, Dec 21 2006 *)

a(2n-1) = A005732(2n-1) for n > 1; a(2n) = A005732(2n) - A260417(n) for n > 1. - Jonathan Sondow, Jul 25 2015

a(3)-a(8) computed by Victor Meally (personal communication to N. J. A. Sloane, circa 1975); later terms and recurrence from S. Sommars and T. Sommars.

A326210 Number of labeled simple graphs with vertices {1..n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.

Original entry on oeis.org

0, 0, 0, 0, 16, 672, 29888, 2071936, 268204288, 68717285888, 35184350796800, 36028796807919616, 73786976292712960000, 302231454903635611721728, 2475880078570760326175178752, 40564819207303340845566684397568, 1329227995784915872903782635437883392

Offset: 0

Gus Wiseman, Jun 19 2019

nonn

Also simple graphs containing a crossing pair of edges, where two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b.

Also simple graphs such that, if the edges are listed in lexicographic order, their maxima (seconds) are not weakly increasing.

			The a(4) = 16 nesting edge-sets:
  {14,23}
  {12,14,23}
  {13,14,23}
  {14,23,24}
  {14,23,34}
  {12,13,14,23}
  {12,14,23,24}
  {12,14,23,34}
  {13,14,23,24}
  {13,14,23,34}
  {14,23,24,34}
  {12,13,14,23,24}
  {12,13,14,23,34}
  {12,14,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
The a(4) = 16 crossing edge-sets:
  {13,24}
  {12,13,24}
  {13,14,24}
  {13,23,24}
  {13,24,34}
  {12,13,14,24}
  {12,13,23,24}
  {12,13,24,34}
  {13,14,23,24}
  {13,14,24,34}
  {13,23,24,34}
  {12,13,14,23,24}
  {12,13,14,24,34}
  {12,13,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Non-nesting graphs are A054726.
Nesting digraphs are A326209.
Nesting (or crossing) set partitions are A016098.
MM-numbers of nesting multiset partitions are A326256.
Cf. A001519, A006125, A095661, A324170, A324172, A324326.
Cf. A326211, A326243, A326244, A326248, A326250.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!OrderedQ[Last/@#]&]],{n,0,5}]

seq(n)={my(p=1 + 3/2*x - x^2 - x/2*sqrt(1 - 12*x + 4*x^2 + O(x^n))); concat([0], vector(n, k, 2^binomial(k,2)-polcoef(p,k)))} \\ Andrew Howroyd, Aug 26 2019

A006125(n) = a(n) + A054726(n).

Terms a(7) and beyond from Andrew Howroyd, Aug 26 2019

A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 1, 4, 39, 748, 27162, 1880872, 252273611, 66358216668, 34506398937158, 35644762692112792, 73356520492898454022, 301274559225693420690360, 2471654510727312089903896948, 40527708183358718551543295827536, 1328579216048284168977214446788083699

Offset: 0

Paul D. Hanna, Jan 15 2008

nonn

a(n) is the number of graphs on vertices 1,...,n such that, when these vertices are arranged counterclockwise around a circle and edges are drawn as straight line segments, the resulting diagram is connected. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

In this interpretation, both intersecting (set theoretically) and crossing (topologically) edges are considered connected. - Gus Wiseman, Feb 23 2019

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 39*x^4 + 748*x^5 + 27162*x^6 +...
Let F(x) = 1 + x + 2*x^2 + 8*x^3 + 64*x^4 + 1024*x^5 +...+ 2^(n*(n-1)/2)*x^n +..
then A(x) = F(x/A(x)), A(x*F(x)) = F(x).
Coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2),
as can be seen by the main diagonal in the array of
coefficients in the initial powers of A(x):
A^1: [(1), 1, 1, 4, 39, 748, 27162, 1880872, 252273611,...;
A^2: [1, (2), 3, 10, 87, 1582, 55914, 3817876, 508370795,...;
A^3: [1, 3, (6), 19, 147, 2517, 86398, 5813550, 768378627,...;
A^4: [1, 4, 10, (32), 223, 3572, 118778, 7870640, 1032387787,...;
A^5: [1, 5, 15, 50, (320), 4771, 153245, 9992130, 1300492845,...;
A^6: [1, 6, 21, 74, 444, (6144), 190023, 12181278, 1572792585,...;
A^7: [1, 7, 28, 105, 602, 7728, (229376), 14441659, 1849390375,...;
A^8: [1, 8, 36, 144, 802, 9568, 271616, (16777216), 2130394591,...;
A^9: [1, 9, 45, 192, 1053, 11718, 317112, 19192320, (2415919104),...;
dividing each diagonal term in row n by (n+1) gives 2^(n*(n-1)/2).
The diagonal above the main diagonal gives coefficients of l.g.f.:
log(F(x)) = x + 3*x^2/2 + 19*x^3/3 + 223*x^4/4 + 4771*x^5/5 +...

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 39 graphs connected by overlaps or crossings.

Cf. A136652 (log(A(x))), A136654.
Cf. A000699, A002662, A006125, A007297, A016098, A054726, A099947, A306438.
Cf. A324166, A324169, A324172, A324173, A324327, A324328.

max = 15; s = x*Sum[2^(k*(k-1)/2)*x^k, {k, 0, max}] + O[x]^(max+2); x/InverseSeries[s] + O[x]^(max+1) // CoefficientList[#, x]& (* Jean-François Alcover, Sep 03 2017 *)
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
bicmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],Intersection@@#!={}&],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[bicmpts[#]]<=1]&]],{n,0,5}] (* Gus Wiseman, Feb 23 2019 *)

a(n)=polcoeff(x/serreverse(x*sum(k=0,n,2^(k*(k-1)/2)*x^k +x*O(x^n))),n)

G.f.: A(x) = x/Series_Reversion( x*Sum_{k=0..n} 2^(k(k-1)/2)*x^k ).

Equals the free cumulant sequence corresponding to A006125. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

Name changed and part of prior name moved to formula section by Paul D. Hanna, Sep 19 2013

A326244 Number of labeled n-vertex simple graphs without crossing or nesting edges.

Original entry on oeis.org

1, 1, 2, 8, 36, 160, 704, 3088, 13536, 59328, 260032, 1139712

Offset: 0

Gus Wiseman, Jun 20 2019

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.

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
Gus Wiseman, The a(4) = 36 non-crossing, non-nesting simple labeled graphs.

The case for set partitions is A001519.
Simple graphs with crossing or nesting edges are A326279.
Cf. A000088, A000108, A006125, A016098, A054726, A095661, A117662.
Cf. A326209, A326210, A326250, A326255, A326256.

croXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x

A006125(n) = a(n) + A326279(n).
Conjectures from Colin Barker, Jun 28 2019: (Start)
G.f.: (1 - x)*(1 - 4*x) / (1 - 6*x + 8*x^2 - 4*x^3).
a(n) = 6*a(n-1) - 8*a(n-2) + 4*a(n-3) for n>2.
(End)

A324167 Number of non-crossing antichain covers of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 9, 67, 633, 6763, 77766, 938957, 11739033, 150649945, 1973059212, 26265513030, 354344889798, 4833929879517, 66568517557803, 924166526830701, 12920482325488761, 181750521972603049, 2570566932237176232, 36532394627404815308, 521439507533582646156

Offset: 0

Gus Wiseman, Feb 17 2019

nonn

An antichain is non-crossing if no pair of distinct parts is of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

			The a(3) = 9 antichains:
  {{1,2,3}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}

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

Cf. A000108, A000124, A000372 (antichains), A001006, A006126 (antichain covers), A014466, A048143, A054726 (non-crossing graphs), A099947, A261005, A283877, A306438.

Cf. A324166, A324168, A324169, A324170, A324171, A324173, A359984 (no singletons).

nn=6;
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

seq(n)={my(f=O(1)); for(n=2, n, f = 1 + (4*x + x^2)*f^2 - 3*x^2*(1 + x)*f^3); Vec(subst(x*(1 + x^2*f^2 - 3*x^3*f^3), x, x/(1-x))/x) } \\ Andrew Howroyd, Jan 20 2023

Inverse binomial transform of A324168.

Binomial transform of A359984. - Andrew Howroyd, Jan 20 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 20 2023

A326293 Number of non-nesting, topologically connected simple graphs with vertices {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 8, 27, 192, 1750

Offset: 0

Gus Wiseman, Jun 29 2019

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. A graph with positive integer vertices is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.

Gus Wiseman, The a(5) = 27 non-nesting, topologically connected simple graphs.

The inverse binomial transform is the covering case A326349.
Topologically connected simple graphs are A324328.
Non-crossing simple graphs are A054726.
Topologically connected set partitions are A099947.
Cf. A000108, A000699, A006125, A007297, A117662, A136653.
Cf. A324173, A324323, A324327, A326244, A326330, A326341.

croXQ[eds_]:=MatchQ[eds,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!nesXQ[#]&&Length[csm[Union[Subsets[#,{1}],Select[Subsets[#,{2}],croXQ]]]]<=1&]],{n,0,5}]

A324166 Number of totally crossing set partitions of {1,...,n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 18, 57, 207, 842, 3673, 17062, 84897

Offset: 0

Gus Wiseman, Feb 17 2019

A set partition is totally crossing if every pair of distinct blocks is of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y.

			The a(6) = 18 totally crossing set partitions:
  {{1,2,3,4,5,6}}
  {{1,4,6},{2,3,5}}
  {{1,4,5},{2,3,6}}
  {{1,3,6},{2,4,5}}
  {{1,3,5},{2,4,6}}
  {{1,3,4},{2,5,6}}
  {{1,2,5},{3,4,6}}
  {{1,2,4},{3,5,6}}
  {{4,6},{1,2,3,5}}
  {{3,6},{1,2,4,5}}
  {{3,5},{1,2,4,6}}
  {{2,6},{1,3,4,5}}
  {{2,5},{1,3,4,6}}
  {{2,4},{1,3,5,6}}
  {{1,5},{2,3,4,6}}
  {{1,4},{2,3,5,6}}
  {{1,3},{2,4,5,6}}
  {{1,4},{2,5},{3,6}}

Cf. A000108 (non-crossing partitions), A000110, A000296, A002662, A016098 (crossing partitions), A054726, A099947 (topologically connected partitions), A305854, A306006, A306418, A306438, A319752.

Cf. A324170, A324171, A324172, A324173.

nn=6;
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

cp's OEIS Frontend

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

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A006533 Place n equally-spaced points around a circle and join every pair of points by a chord; this divides the circle into a(n) regions.

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A324173 Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.

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A006600 Total number of triangles visible in regular n-gon with all diagonals drawn.

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A326210 Number of labeled simple graphs with vertices {1..n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.

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A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

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A326244 Number of labeled n-vertex simple graphs without crossing or nesting edges.

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