A291841 -id:A291841 - cp's OEIS frontend

A291842 a(n) is the number of labeled connected planar graphs with n edges.

1, 3, 17, 140, 1524, 20673, 336259, 6382302, 138525770, 3384987698, 91976075664, 2751117418712, 89832957177685, 3179833729806525, 121286809954760876, 4959277317653328656, 216402696660205555698, 10037527922988058277877, 493159461152794975438450, 25585023231409205439510792

Offset: 1

Gheorghe Coserea, Sep 10 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..123
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
P. S. Kolesnikov and B. K. Sartayev, On the special identities of Gelfand--Dorfman algebras, arXiv:2105.13815 [math.RA], 2021.

Cf. A096332, A290326, A291841.

Column sums of A288265.

Q(n,k) = { \\ c-nets with n-edges, k-vertices
  if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
  sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
  (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
  4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
};
seq(N) = {
my(x='x+O('x^(N+3)), t='t,
   q=t*x*Ser(vector(N, n, Polrev(vector(2*n\3, k, Q(n,k)),t))),
   d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
   g2=intformal(t^2/2*((1+d)/(1+x)-1)),
   b=t*'x^2/2 + 'x*Ser(vector(N+1, n, subst(polcoeff(g2, n, 't),'x,'t))),
   g1=intformal(serreverse('x/exp(b'))/'x),
   e1='x*Ser(vector(N, n, subst(polcoeff(serlaplace(g1), n, 't), 'x, 't))));
   Vec(subst(e1,'t,1));
};
seq(20)

A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 0, 1, 3, 18, 131, 1180, 12570, 154535, 2151439, 33431046, 573197723, 10743619285, 218447494812, 4787255999220, 112454930390211, 2818138438707516, 75031660452368001, 2114705500316025737, 62890323682634277951, 1967901134191778583146, 64623905086814216468839

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100
Gus Wiseman, The a(6) = 131 labeled 2-connected graphs with 6 edges.

Cf. A002218, A013922, A123534, A275307, A291841, A304118, A304887, A322110, A322117, A322118, A322137, A322138.

seq(n)={Vec(1 + vecsum(Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2) * x^k / k!) + O(x*x^n)))))))))} \\ Andrew Howroyd, Nov 29 2018

a(n) = Sum_{i=3..n} A123534(i, n). - Andrew Howroyd, Nov 30 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 29 2018

A343869 Number of unlabeled nonseparable (or 2-connected) planar graphs with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 7, 16, 41, 108, 320, 1042, 3575, 13064, 49938, 197729, 805991, 3363084, 14302891, 61813285, 270805177, 1200460492, 5376709415, 24302430375, 110745093999, 508380790741

Offset: 1

Andrew Howroyd, May 04 2021

Terms may be computed using the tools geng and planarg in nauty.

Brendan McKay and Adolfo Piperno, nauty and Traces

Row sums of A343870.
Column sums of A049336(n > 1).
Cf. A002840 (3-connected), A010355, A021103, A046091, A289471, A291841.

# count graphs for the sequence by number of vertices v, sum over v afterwards
geng -C $v $n:$n | planarg -q | countg -q # Georg Grasegger, Jun 05 2023

a(21)-a(26) added by Georg Grasegger, Jun 05 2023

A322140 Number of labeled 2-connected multigraphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 1, 2, 7, 37, 262, 2312, 24338, 296928, 4112957, 63692909, 1089526922, 20389411551, 414146189901, 9070116944468, 212983762029683, 5336570227705763, 142083405456873290, 4004953714929148655, 119128974685786590410, 3728639072095285867881

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

We consider a single edge to be 2-connected, so a(1) = 1.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Gus Wiseman, The a(5) = 37 labeled 2-connected multigraphs with 5 edges.

Cf. A002218, A002905, A013922, A275307, A291841, A304118, A304887, A322110, A322117, A322118, A322137, A322138.

seq(n)={Vec(1 + vecsum(Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, 1/(1 - y + O(y*y^n))^binomial(k, 2) * x^k / k!) + O(x*x^n)))))))))} \\ Andrew Howroyd, Nov 29 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 29 2018

A343871 Number of labeled 3-connected planar graphs with n edges.

Original entry on oeis.org

1, 0, 15, 70, 432, 4320, 30855, 294840, 2883240, 28175952, 310690800, 3458941920, 40459730640, 499638948480, 6324655705200, 83653192972800, 1145266802114400, 16145338385736000, 235579813593453000, 3535776409508703360, 54571687068401395200, 866268656574795936000

Offset: 6

Andrew Howroyd, May 05 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 6..200
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, pp. 377-386.

Cf. A000287, A002840 (unlabeled case), A096330, A290326, A291841, A338414.

Q(n,k) = { \\ c-nets with n-edges, k-vertices (see A290326)
  if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
  sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
  (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
  4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
};
a(n)={sum(k=2+(n+2)\3, 2*n\3, k!*Q(n,k))/(4*n)} \\ Andrew Howroyd, May 05 2021

a(n) = Sum_{k=2+floor((n+2)/3)..floor(2*n/3)} k!*A290326(n-k+1, k-1)/(4*n).

cp's OEIS Frontend

A291842 a(n) is the number of labeled connected planar graphs with n edges.

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A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

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PARI

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A343869 Number of unlabeled nonseparable (or 2-connected) planar graphs with n edges.

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nauty

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A322140 Number of labeled 2-connected multigraphs with n edges (the vertices are {1,2,...,k} for some k).

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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A343871 Number of labeled 3-connected planar graphs with n edges.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula