A005142 -id:A005142 - cp's OEIS frontend

A033995 Number of bipartite graphs with n nodes.

1, 1, 2, 3, 7, 13, 35, 88, 303, 1119, 5479, 32303, 251135, 2527712, 33985853, 611846940, 14864650924, 488222721992, 21712049275198, 1308300679611469, 106897965189674291, 11852113048215107822, 1784730721403509209215, 365323537513403184463273

Offset: 0

Ronald C. Read

All bipartite graphs are perfect. - Falk Hüffner, Nov 27 2015

EULER transform of A005142 [1, 1, 1, 3, 5, 17, ...] is [1, 2, 3, 7, 13, ...]. - Michael Somos, May 13 2019

			For n=1: o; n=2: o o, o-o; n=3: o o o, o o-o, o-o-o; n=4: o o o o, o o o-o, o-o o-o, o o-o-o, o-o-o-o, K_{2,2}, K_{3,1}. - _Michael Somos_, May 13 2019

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

Andrew Howroyd, Table of n, a(n) for n = 0..50
CombOS - Combinatorial Object Server, Generate graphs.
P. Erdős, D. J. Kleitman, and B. L. Rothschild, Asymptotic enumeration of k_n-free graphs. In Colloquio Internazionale sulle Teorie Combinatorie, (Rome, 1973), Tomo II, Atti dei Convegni Lincei, No. 17, pp. 19-27. Accad. Naz. Lincei, Rome.
P. Hanlon, The enumeration of bipartite graphs, Discrete Math. 28 (1979), 49-57.
S. Hougardy, Home Page.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
Sage, Common Graphs (Graph Generators).
Eric Weisstein's World of Mathematics, Bipartite Graph.
Eric Weisstein's World of Mathematics, Bicolorable Graph.
Eric Weisstein's World of Mathematics, n-Colorable Graph.

Row sums of A297877.
The labeled version is A047864.
Equals A076278(n) + 1.
Cf. A005142 (connected).

A005142 = Cases[Import["https://oeis.org/A005142/b005142.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A005142] (* Jean-François Alcover, Mar 18 2020 *)

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Sep 05 2018

A001832 Number of labeled connected bipartite graphs on n nodes.

Original entry on oeis.org

1, 1, 3, 19, 195, 3031, 67263, 2086099, 89224635, 5254054111, 426609529863, 47982981969979, 7507894696005795, 1641072554263066471, 502596525992239961103, 216218525837808950623459, 130887167385831881114006475, 111653218763166828863141636911

Offset: 1

N. J. A. Sloane

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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..50
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy)
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
A. Nymeyer and R. W. Robinson, Tabulation of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]
Eric Weisstein's World of Mathematics, n-Colorable Graph
Eric Weisstein's World of Mathematics, n-Chromatic Graph

Row sums of A228861.
The unlabeled version is A005142.
Cf. A002031, A047863, A047864.

mx = 17; s = Sum[ Binomial[n, k] 2^(k (n - k)) x^n/n!, {n, 0, mx}, {k, 0, n}] ; Range[0, mx]! CoefficientList[ Series[ Log[s]/2, {x, 0, mx}], x] (* Geoffrey Critzer, May 10 2011 *)

seq(n)=Vec(serlaplace(log(sum(k=0, n, exp(2^k*x + O(x*x^n))*x^k/k!))/2)) \\ Andrew Howroyd, Sep 26 2018

E.g.f.: log(A(x))/2 where A(x) is e.g.f. of A047863.

a(n) = A002031(n)/2, for n > 1. - Geoffrey Critzer, May 10 2011

More terms from Vladeta Jovovic, Apr 12 2003

A076278 Number of 2-chromatic (i.e., chromatic number equals 2) simple graphs on n nodes.

Original entry on oeis.org

0, 1, 2, 6, 12, 34, 87, 302, 1118, 5478, 32302, 251134, 2527711, 33985852, 611846939, 14864650923, 488222721991, 21712049275197, 1308300679611468, 106897965189674290, 11852113048215107821, 1784730721403509209214, 365323537513403184463272

Offset: 1

Eric W. Weisstein, Oct 06 2002

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Keith M. Briggs, Combinatorial Graph Theory
Eric Weisstein's World of Mathematics, n-Chromatic Graph

Column k=2 of A084268.

Cf. A076279, A076280, A076281, A076282, A115597.

A005142 = Import["https://oeis.org/A005142/b005142.txt", "Table"][[All, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
a = etr[A005142[[# + 1]]&][#] - 1&;
Array[a, 23] (* Jean-François Alcover, Sep 03 2019 *)

a(n) = A033995(n)-1.

More terms from Vladeta Jovovic, Jul 31 2003

Terms a(21) and beyond from Andrew Howroyd, Sep 05 2018

A084269 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 5, 12, 3, 1, 0, 17, 64, 26, 4, 1, 0, 44, 475, 282, 46, 5, 1, 0, 182, 5036, 5009, 809, 74, 6, 1, 0, 730, 80947, 149551, 27794, 1940, 110, 7, 1, 0, 4032, 2010328, 7694428, 1890221, 113272, 4125, 156, 8, 1, 0, 25598, 76115143, 667036310, 248580644, 14545025, 389583, 8040, 212, 9, 1

Offset: 0

Eric W. Weisstein, May 24 2003

			Triangle begins:
  1;
  0,   1;
  0,   1,     1;
  0,   3,     2,      1;
  0,   5,    12,      3,     1;
  0,  17,    64,     26,     4,    1;
  0,  44,   475,    282,    46,    5,   1;
  0, 182,  5036,   5009,   809,   74,   6, 1;
  0, 730, 80947, 149551, 27794, 1940, 110, 7, 1;
  ...

Eric Weisstein's World of Mathematics, n-Chromatic Graph

Row sums are A001349.
Columns k=3..7 are A126737, A126738, A126739, A126740, A241702.
Partial row sums include A005142, A076322, A076323, A076324, A076325, A076326, A076327, A076328.
Essentially the same table as A126736.
Cf. A084268 (not necessarily connected), A115597.

a(37)-a(66) from Andrew Howroyd, Dec 02 2018

A007776 Number of connected posets with n elements of height 1.

Original entry on oeis.org

1, 2, 4, 10, 27, 88, 328, 1460, 7799, 51196, 422521, 4483460, 62330116, 1150504224, 28434624153, 945480850638, 42417674401330, 2572198227615998, 211135833162079184, 23487811567341121158, 3545543330739039981738, 727053904070651775719646

Offset: 2

Georg Wambach (gw(AT)informatik.Uni-Koeln.de)

nonn

Inverse Euler transform of A048194 and A049312. - Detlef Pauly (dettodet(AT)yahoo.de) and Vladeta Jovovic, Jul 25 2003
Essentially the same as A318870. - Georg Fischer, Oct 02 2018
Number of connected digraphs on n unlabeled nodes where every node has indegree 0 or outdegree 0 and there are no isolated nodes.  - Andrew Howroyd, Oct 03 2018

Andrew Howroyd, Table of n, a(n) for n = 2..50 (terms 2..40 from Alois P. Heinz)
N. J. A. Sloane, Transforms
J. Textor, A. Idelberger, and M. Liskiewicz, Learning from Pairwise Marginal Independencies, arXiv:1508.00280 [cs.AI], 2015.
Index entries for sequences related to posets

Cf. A005142, A002031 (labeled case), A048194, A049312, A055192, A318870, column 1 of A342500.

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
EULERi[Array[b, 30]] // Rest (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A049312 *)

Inverse Euler transform of A055192. - Andrew Howroyd, Oct 03 2018

More terms from Vladeta Jovovic, Jul 25 2003

Offset corrected by Andrew Howroyd, Oct 03 2018

A076322 Number of connected 3-colorable (i.e., chromatic number <= 3) simple graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 5, 17, 81, 519, 5218, 81677, 2014360, 76140741, 4303246908

Offset: 1

Eric W. Weisstein, Oct 06 2002

Maria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, Mingxian Zhong, Obstructions for three-coloring graphs without induced paths on six vertices, arXiv preprint, arXiv:1504.06979 [math.CO], 2015-2018.
Eric Weisstein's World of Mathematics, n-Colorable Graph

Cf. A005142, A076323, A076324, A076325, A076326, A076327, A076328.

Cf. A076279, A076315, A084269, A126737.

A076315 = Cases[Import["https://oeis.org/A076315/b076315.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[A076315] (* Jean-François Alcover, Sep 25 2019, updated Mar 17 2020 *)

Inverse Euler transform of A076315. - Andrew Howroyd, Dec 02 2018

a(10)-a(11) from Andrew Howroyd, Dec 02 2018

a(12) from Jinyuan Wang, Feb 23 2020

A034889 Number of embeddings on the sphere of 2-connected planar graphs with n nodes.

Original entry on oeis.org

1, 3, 10, 61, 564, 7593, 123874, 2262877, 44190279, 904777809, 19207129217, 419870351012, 9405626692325

Offset: 3

Ronald C. Read

The complete graph on two vertices is sometimes considered to be 2-connected (or nonseparable). Compare A002218 with A021103. - Andrew Howroyd, Mar 01 2021

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem, 58 (2007), 323-357. [Expanded version]
Eric Weisstein's World of Mathematics, Planar Embedding.

Row sums of A342060.

Cf. A002218, A005142, A021103.

a(8)-a(15) added by Mohammadreza Jooyandeh, Sep 03 2013

A076323 Number of connected 4-colorable (i.e., chromatic number <= 4) simple graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 6, 20, 107, 801, 10227, 231228, 9708788, 743177051, 100580560531

Offset: 1

Eric W. Weisstein, Oct 06 2002

Maria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, Mingxian Zhong, Obstructions for three-coloring graphs without induced paths on six vertices, arXiv preprint, arXiv:1504.06979 [math.CO], 2015-2018.
Eric Weisstein's World of Mathematics, n-Colorable Graph

Cf. A005142, A076322, A076324, A076325, A076326, A076327, A076328.

Cf. A076280, A076316, A084269, A126738.

A076316 = Cases[Import["https://oeis.org/A076316/b076316.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[A076316] (* Jean-François Alcover, Sep 25 2019, updated Mar 17 2020 *)

Inverse Euler transform of A076316. - Andrew Howroyd, Dec 02 2018

a(10)-a(11) from Andrew Howroyd, Dec 02 2018

a(12) from Sean A. Irvine, Apr 13 2025

A076324 Number of connected 5-colorable (i.e., chromatic number <= 5) simple graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 6, 21, 111, 847, 11036, 259022, 11599009, 991757695

Offset: 1

Eric W. Weisstein, Oct 06 2002

Eric Weisstein's World of Mathematics, k-Colorable Graph

Cf. A005142, A076322, A076323, A076325, A076326, A076327, A076328.

Cf. A076281, A076317, A084269, A126739.

Inverse Euler transform of A076317. - Andrew Howroyd, Dec 02 2018

a(10)-a(11) from Andrew Howroyd, Dec 02 2018

A076325 Number of connected 6-colorable (i.e., chromatic number <= 6) simple graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 6, 21, 112, 852, 11110, 260962, 11712281, 1006302720

Offset: 1

Eric W. Weisstein, Oct 06 2002

Eric Weisstein's World of Mathematics, n-Colorable Graph

Cf. A005142, A076322, A076323, A076324, A076326, A076327, A076328.

Cf. A076282, A076318, A084269, A126740.

Inverse Euler transform of A076318. - Andrew Howroyd, Dec 02 2018

a(10)-a(11) from Andrew Howroyd, Dec 02 2018

cp's OEIS Frontend

A033995 Number of bipartite graphs with n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A001832 Number of labeled connected bipartite graphs on n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A076278 Number of 2-chromatic (i.e., chromatic number equals 2) simple graphs on n nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A084269 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A007776 Number of connected posets with n elements of height 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A076322 Number of connected 3-colorable (i.e., chromatic number <= 3) simple graphs on n nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A034889 Number of embeddings on the sphere of 2-connected planar graphs with n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A076323 Number of connected 4-colorable (i.e., chromatic number <= 4) simple graphs on n nodes.

Views

Author

Keywords

Links

Crossrefs

Programs