A100960 -id:A100960 - cp's OEIS frontend

A291837 a(n) is the maximal value in row n of triangle A100960.

1, 6, 100, 3525, 210861, 20545920, 2516883516, 366723015750, 65231311386780, 13434052797314820, 3068032280097740670, 770387691039763211415, 222066633621598291951425, 69102739152239837029025040, 23037728813031184811224116360

Offset: 3

Gheorghe Coserea, Sep 04 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 3..126
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.

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))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
   q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 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)));
   serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
};
N=15; apply(p->vecmax(Vecrev(p)), Vec(A100960_ser(N+2)))

a(n) ~ k * n^(-4) * r^n * n!, where k=0.000002389700772064... (A291838) and r=26.1841125556... (A291836) (see Bender link).

A291839 a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A100960.

Original entry on oeis.org

3, 5, 7, 9, 11, 14, 16, 18, 21, 23, 25, 27, 30, 32, 34, 37, 39, 41, 43, 46, 48, 50, 52, 55, 57, 59, 61, 64, 66, 68, 71, 73, 75, 77, 80, 82, 84, 86, 89, 91, 93, 95, 98, 100, 102, 104, 107, 109, 111, 114, 116, 118, 120, 123, 125, 127, 129, 132

Offset: 3

Gheorghe Coserea, Sep 05 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 3..126
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.

Cf. A100960, A291837.

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))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
   q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 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)));
   serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
};
A291839_seq(N) = {
  my(g2=apply(Vecrev, Vec(A100960_ser(N+2))), m=apply(vecmax, g2));
  apply(v->vecmin(v)-1, vector(#g2, k, select(v->v==m[k], g2[k], 1)));
};
A291839_seq(22)

a(n) ~ c*n + o(sqrt(n)), where c=2.26287583256262... (A291840).

T(n, a(n)) = max {T(n,k), n <= k <= 3*(n-2) }, where T(n,k) is defined by A100960.

A096331 Number of 2-connected planar graphs on n labeled nodes.

Original entry on oeis.org

1, 10, 237, 10707, 774924, 78702536, 10273189176, 1631331753120, 304206135619160, 65030138045062272, 15659855107404275280, 4191800375194003211360, 1234179902360142341550240, 396280329098426228719121280, 137779269467538258010671193472

Offset: 3

Steven Finch, Aug 02 2004

nonn

Recurrence known, see Bodirsky et al.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.

Gheorghe Coserea, Table of n, a(n) for n = 3..126
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, ICALP03 Eindhoven, LNCS 2719, Springer Verlag (2003), 1095 - 1107.
O. Gimenez and M. Noy, Asymptotic enumeration and limit laws of planar graphs, arXiv:math/0501269 [math.CO], 2005.

Cf. A066537. Row sums of A100960.

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))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
   q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 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)));
   serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
};
Vec(subst(A100960_ser(20),'t,1)) \\ Gheorghe Coserea, Aug 10 2017

a(n) ~ g * n^(-7/2) * r^n * n!, where g=0.00000370445941594... (A291835) and r=26.1841125556... (A291836) (see Bender link). - Gheorghe Coserea, Sep 03 2017

More terms from Gheorghe Coserea, Aug 05 2017

A066537 Number of planar graphs on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 8, 64, 1023, 32071, 1823707, 163947848, 20402420291, 3209997749284, 604611323732576, 131861300077834966, 32577569614176693919, 8977083127683999891824, 2726955513946123452637877, 904755724004585279250537376, 325403988657293080813790670641

Offset: 0

Aart Blokhuis (aartb(AT)win.tue.nl), Jan 08 2002

Precise numbers derived from numbers of 3-connected, 2-connected and 1-connected planar labeled graphs. Details and more entries in Bodirsky et al. Some bounds on the asymptotics are known, see e.g. Taraz et al.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.

Keith M. Briggs and Gheorghe Coserea, Table of n, a(n) for n = 0..126, terms 0..42 from Keith M. Briggs.
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, ICALP03 Eindhoven, LNCS 2719, Springer Verlag (2003), 1095 - 1107.
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
Keith M. Briggs, Combinatorial Graph Theory
O. Gimenez and M. Noy, Asymptotic enumeration and limit laws of planar graphs, arXiv:math/0501269 [math.CO], 2005.
Yu Nakahata, Jun Kawahara, Takashi Horiyama, Shin-ichi Minato, Implicit Enumeration of Topological-Minor-Embeddings and Its Application to Planar Subgraph Enumeration, arXiv:1911.07465 [cs.DS], 2019.
A. Taraz, D. Osthus and H. J. Proemel, On random planar graphs, the number of planar graphs and their triangulations Journal of Combinatorial Theory, Series B, 88 (2003), 119-134.

Cf. A005470, A096330, A096331, A096332 (connected), A100960, A266390, A266391.

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))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
   q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 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)));
   serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
};
A096331_seq(N) = Vec(subst(A100960_ser(N+2),'t,1));
A096332_seq(N) = {
  my(x='x+O('x^(N+3)), b=x^2/2+serconvol(Ser(A096331_seq(N))*x^3, exp(x)));
  Vec(serlaplace(intformal(serreverse(x/exp(b'))/x)));
};
A066537_seq(N) = {
  my(x='x+O('x^(N+3)));
  Vec(serlaplace(exp(serconvol(Ser(A096332_seq(N))*'x,exp(x)))));
};
A066537_seq(15) \\ Gheorghe Coserea, Aug 10 2017

Recurrence known, see Bodirsky et al.

a(n) ~ g * n^(-7/2) * gamma^n * n!, where g=0.000004260938569161439...(A266391) and gamma=27.2268777685...(A266390) (see Gimenez and Noy).

More terms from Manuel Bodirsky (bodirsky(AT)informatik.hu-berlin.de), Sep 15 2003

Entry revised by N. J. A. Sloane, Jun 17 2006

A291841 a(n) is the number of labeled 2-connected planar graphs with n edges.

Original entry on oeis.org

1, 3, 18, 131, 1180, 12570, 154525, 2150748, 33399546, 571979428, 10699844995, 216921707622, 4734437392728, 110613829184421, 2752971531611715, 72676980383698345, 2027560176161932735, 59579981648921326791, 1838669555339295257097, 59435431024069408426431

Offset: 3

Gheorghe Coserea, Sep 10 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 3..123
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.

Cf. A096331, A290326.

Column sums of A100960.

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)), e2=apply(serlaplace, g2));
   Vec(subst(e2, 't, 1));
};
seq(22)

A288265 Triangle read by rows: T(n,k) is the number of labeled connected planar graphs on n vertices and k edges.

Original entry on oeis.org

1, 1, 3, 1, 16, 15, 6, 1, 125, 222, 205, 120, 45, 10, 1296, 3660, 5700, 6165, 4935, 2937, 1125, 195, 16807, 68295, 156555, 258125, 330456, 334530, 254275, 131985, 40950, 5712, 262144, 1436568, 4483360, 10230360, 18528216, 27261192, 31761744, 27958920, 17666320, 7513632, 1922760, 223440, 4782969, 33779340, 136368414, 405918324, 970196283, 1910996136, 3058785990, 3866563764, 3754432899, 2724326136, 1425385584, 507370500, 109907280, 10929600

Offset: 1

Gheorghe Coserea, Aug 14 2017

Row n >= 3 contains 3*n-5 terms.

			A(x;t) = x + t*x^2/2! + (3*t^2 + t^3)*x^3/3! + (16*t^3 + 15*t^4 + 6*t^5 + t^6)*x^4/4! + ...
Triangle starts:
n\k [0] [1] [2] [3] [4]  [5]   [6]   [7]   [8]   [9]   [10]  [11]  [12]
[1] 1;
[2] 0   1;
[3] 0,  0,  3,  1;
[4] 0,  0,  0,  16, 15,  6,    1;
[5] 0,  0,  0,  0,  125, 222,  205,  120,  45,   10;
[6] 0,  0,  0,  0,  0,   1296, 3660, 5700, 6165, 4935, 2937, 1125, 195;
[7] ...

Gheorghe Coserea, Rows n = 1..126, flattened
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
Omer Gimenez, Marc Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. 22 (2009), 309-329.

Cf. A000272, A007816, A096332, A100960, A266390, A267411, A288266, A290326.

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))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
   q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 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)));
   serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
};
A288265_ser(N) = {
  my(x='x+O('x^(N+3)), b = t*x^2/2 + serconvol(A100960_ser(N), exp(x)),
     g1=intformal(serreverse(x/exp(b'))/x)); serlaplace(g1);
};
A288265_seq(N) = {
  my(v=Vec(A288265_ser(N))); vector(#v, n, Vecrev(v[n]/t^(n-1)));
};
concat(A288265_seq(9))

A096332(n) = Sum_{k=n-1..3*n-6} T(n,k) for n >= 3.

A000272(n) = T(n,n-1), A007816(n-3) = T(n, 3*n-6).

A288266 Triangle read by rows: T(n,k) is the number of labeled planar graphs on n vertices and k edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 15, 20, 15, 6, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 4995, 2937, 1125, 195, 1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293860, 351225, 342405, 255640, 131985, 40950, 5712, 1, 28, 378, 3276, 20475, 98280, 376740, 1184040, 3108105, 6906620, 13112694, 21322812, 29332947, 32823084, 28286520, 17712016, 7513632, 1922760, 223440

Offset: 0

Gheorghe Coserea, Aug 14 2017

Row n >= 3 contains 3*n-5 terms.

			A(x;t) = 1 + x + (1+t)*x^2/2! + (1+3*t+3*t^2+t^3)*x^3/3! + (1+6*t+15*t^2+20*t^3+15*t^4+6*t^5+t^6)*x^4/4! + ...
Triangle starts:
n\k [0] [1] [2]  [3]  [4]   [5]   [6]   [7]   [8]   [9]   [10]  [11]  [12]
[0] 1;
[1] 1;
[2] 1   1;
[3] 1,  3,  3,   1;
[4] 1,  6,  15,  20,  15,   6,    1;
[5] 1,  10, 45,  120, 210,  252,  210,  120,  45,   10;
[6] 1,  15, 105, 455, 1365, 3003, 5005, 6435, 6435, 4995, 2937, 1125, 195;
[7] ...

Gheorghe Coserea, Rows n = 0..126, flattened
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
Omer Gimenez, Marc Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. 22 (2009), 309-329.

Cf. A096332, A100960, A266390, A288265, A290326.

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))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
   q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 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)));
   serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x);
};
A288266_seq(N) = {
  my(x='x+O('x^(N+3)), b=t*x^2/2 + serconvol(A100960_ser(N), exp(x)),
     g1=intformal(serreverse(x/exp(b'))/x));
  apply(p->Vecrev(p), Vec(serlaplace(exp(g1))));
};
concat(A288266_seq(8))

A066537(n) = Sum_{k=0..3*n-6} T(n,k) for n >= 3.

A007816(n-3) = T(n, 3*n-6).

A291840 Decimal expansion of the constant c in the asymptotic formula for A291839.

Original entry on oeis.org

2, 2, 6, 2, 8, 7, 5, 8, 3, 2, 5, 6, 2, 6, 2, 1, 2, 4, 6, 3, 0, 2, 3, 3, 3, 3, 5, 8, 3, 8, 4, 3, 6, 5, 9, 3, 8, 9, 0, 6, 8, 0, 4, 1, 9, 6, 3, 9, 5, 3, 7, 1, 0, 5, 2, 7, 1, 2, 7, 1, 6, 3, 3, 4, 1, 8, 5, 4, 7, 3, 8, 9, 7, 1, 2, 9, 9, 4, 8

Offset: 1

Gheorghe Coserea, Sep 05 2017

			2.262875832562621246302333358384...

Gheorghe Coserea, Table of n, a(n) for n = 1..55000
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.

Cf. A100960, A266389, A291839.

x(t)     = (1+3*t)*(1/t-1)^3/16;
y(t)     = {
  my(y1  = t^2 * (1-t) * (18 + 36*t + 5*t^2),
     y2  = 2 * (3+t) * (1+2*t) * (1+3*t)^2);
  (1+2*t)/((1+3*t) * (1-t)) * exp(-y1/y2) - 1;
};
alpha(t) = 144 + 592*t + 664*t^2 + 135*t^3 + 6*t^4 - 5*t^5;
mu(t)    = {
  my(mu1 = (1+t) * (3+t)^2 * (1+2*t)^2 * (1+3*t)^2 / t^3, y0 = y(t));
  mu1 * y0 / ((1 + y0) * alpha(t));
};
N=79; default(realprecision, N+100); t0 = solve(t=.62, .63, y(t)-1);
c=mu(t0); eval(select(x->(x != "."), Vec(Str(c))[1..-101]))

Equals mu(A266389), where function t->mu(t) is defined in the PARI code.

Constant c where A291839(n) ~ c*n + o(sqrt(n)).

cp's OEIS Frontend

A291837 a(n) is the maximal value in row n of triangle A100960.

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A291839 a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A100960.

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A096331 Number of 2-connected planar graphs on n labeled nodes.

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A066537 Number of planar graphs on n labeled nodes.

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A291841 a(n) is the number of labeled 2-connected planar graphs with n edges.

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A288265 Triangle read by rows: T(n,k) is the number of labeled connected planar graphs on n vertices and k edges.

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A288266 Triangle read by rows: T(n,k) is the number of labeled planar graphs on n vertices and k edges.

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A291840 Decimal expansion of the constant c in the asymptotic formula for A291839.

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