A001832 -id:A001832 - cp's OEIS frontend

A084284 Number of connected labeled 4-colorable (i.e., chromatic number <= 4) graphs on n nodes.

Original entry on oeis.org

1, 1, 4, 38, 727, 26538, 1832900, 241187797, 60807843592, 29307734588670, 26725594885115575

Offset: 1

Eric W. Weisstein, May 25 2003

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

Cf. A001832, A084274, A084280, A084283, A084285, A084286.

Logarithmic transform of A084280. - Andrew Howroyd, Dec 02 2018

a(7)-a(11) from Andrew Howroyd, Dec 02 2018

A084285 Number of connected labeled 5-colorable (i.e., chromatic number <= 5) graphs on n nodes.

Original entry on oeis.org

1, 1, 4, 38, 728, 26703, 1865842, 251349147, 66151135747, 34319358639795, 35257449916868364

Offset: 1

Eric W. Weisstein, May 25 2003

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

Cf. A001832, A084281, A084283, A084284, A084286.

Logarithmic transform of A084281. - Andrew Howroyd, Dec 02 2018

a(7)-a(11) from Andrew Howroyd, Dec 02 2018

A084286 Number of connected labeled 6-colorable (i.e., chromatic number <= 6) graphs on n nodes.

Original entry on oeis.org

1, 1, 4, 38, 728, 26704, 1866255, 251547611, 66295198837, 34494681184926, 35636712584655663

Offset: 1

Eric W. Weisstein, May 25 2003

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

Cf. A001832, A084282, A084283, A084284, A084285.

Logarithmic transform of A084282. - Andrew Howroyd, Dec 02 2018

a(7)-a(11) from Andrew Howroyd, Dec 02 2018

A322330 Number of 3-colored connected graphs on n labeled nodes up to permutation of the colors.

Original entry on oeis.org

4, 84, 2470, 108390, 7192444, 726782784, 112795368970, 27132558531330, 10196751602156944, 6022337098348167564, 5612248139616665602510, 8274349264629020203315230, 19333678744046195877906230404, 71675537050405087142116150917624, 421915518251999125756688245906168690

Offset: 3

Andrew Howroyd, Dec 03 2018

nonn

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with 3 parts. See A322064 for the definition of stable partition.

Andrew Howroyd, Table of n, a(n) for n = 3..50

Column k=3 of A322278.
Cf. A058873 (not necessarily connected), A322064.
Cf. A001832, A322331.

\\ See A322278 for M.
{ my(N=20); M(N,3)[3..N, 3]~ }

A322331 Number of 4-colored connected graphs on n labeled nodes up to permutation of the colors.

Original entry on oeis.org

38, 3140, 307390, 42747460, 9030799218, 3012940879620, 1628920258500230, 1451200592494754420, 2152262350514389189978, 5344908165470797467243700, 22297912999366719508496874990, 156537595118740106754291705258180, 1850935702258755131781978373277937218

Offset: 4

Andrew Howroyd, Dec 03 2018

nonn

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with 4 parts. See A322064 for the definition of stable partition.

Andrew Howroyd, Table of n, a(n) for n = 4..50

Column k=4 of A322278.
Cf. A058873 (not necessarily connected), A322064.
Cf. A001832, A322330.

\\ See A322278 for M.
{ my(N=20); M(N,4)[4..N, 4]~ }

A084273 Number of labeled connected 3-chromatic (i.e., chromatic number = 3) graphs on n nodes.

Original entry on oeis.org

0, 0, 1, 18, 472, 18855

Offset: 1

Eric W. Weisstein, May 24 2003

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

Cf. A001832, A084274.

A084274 Number of labeled connected 4-chromatic (i.e., chromatic number = 4) graphs on n nodes.

Original entry on oeis.org

0, 0, 0, 1, 60, 4652

Offset: 1

Eric W. Weisstein, May 24 2003

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

Cf. A001832, A084273.

A228859 Triangular array read by rows. T(n,k) is the number of labeled bipartite graphs on n nodes having exactly k connected components; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 19, 15, 6, 1, 195, 125, 45, 10, 1, 3031, 1545, 480, 105, 15, 1, 67263, 27307, 7035, 1400, 210, 21, 1, 2086099, 668367, 140098, 24045, 3430, 378, 28, 1, 89224635, 22427001, 3746925, 536214, 68355, 7434, 630, 36, 1

Offset: 1

Geoffrey Critzer, Sep 05 2013

The Bell transform of A001832(n+1) (without column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016

			1,
1, 1,
3, 3, 1,
19, 15, 6, 1,
195, 125, 45, 10, 1,
3031, 1545, 480, 105, 15, 1,

Row sums are A047864.
Column 1 is A001832.
Cf. A047863.

nn=9;f[x_]:=Sum[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Drop[Range[0,nn]!CoefficientList[Series[Exp[y Log[f[x]]/2],{x,0,nn}],{x,y}],1]]//Grid

# uses[bell_matrix from A264428, A001832]
# Adds 1,0,0,0,... as column 0 to the triangle.
bell_matrix(lambda n: A001832(n+1), 8) # Peter Luschny, Jan 21 2016

E.g.f.: sqrt(A(x)^y) where A(x) is the e.g.f. for A047863.

Sum_{k=1..n} T(n,k)*2^k = A047863(n).

A228861 Irregular triangular array read by rows. T(n,k) is the number of connected labeled bipartite graphs on n nodes with exactly k edges; n >= 1, 0 <= k <= A002620(n+1).

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 0, 0, 0, 16, 3, 0, 0, 0, 0, 125, 60, 10, 0, 0, 0, 0, 0, 1296, 1140, 480, 105, 10, 0, 0, 0, 0, 0, 0, 16807, 23100, 16800, 7770, 2331, 420, 35, 0, 0, 0, 0, 0, 0, 0, 262144, 513240, 555520, 412440, 222936, 88648, 25480, 5040, 616, 35

Offset: 1

Geoffrey Critzer, Sep 05 2013

			Irregular Triangle begins:
  1;
  0, 1;
  0, 0, 3;
  0, 0, 0, 16,   3;
  0, 0, 0,  0, 125,   60,   10;
  0, 0, 0,  0,   0, 1296, 1140, 480, 105, 10;
  ...

Row sums are A001832.

Cf. A002620, A228890.

nn=8;f[x_,y_]:=Sum[Sum[Binomial[n,k](1+y)^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Table[PadLeft[a=Map[Select[#,#>0&]&,Drop[Range[0,nn]!CoefficientList[Series[Log[f[x,y]]/2,{x,0,nn}],{x,y}],1]][[n]],Length[a]+n-1],{n,1,nn}]//Grid

E.g.f.: log(A(x,y))/2 where A(x,y) is the e.g.f. for A228890.

cp's OEIS Frontend

A084284 Number of connected labeled 4-colorable (i.e., chromatic number <= 4) graphs on n nodes.

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A084285 Number of connected labeled 5-colorable (i.e., chromatic number <= 5) graphs on n nodes.

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A084286 Number of connected labeled 6-colorable (i.e., chromatic number <= 6) graphs on n nodes.

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A322330 Number of 3-colored connected graphs on n labeled nodes up to permutation of the colors.

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PARI

A322331 Number of 4-colored connected graphs on n labeled nodes up to permutation of the colors.

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PARI

A084273 Number of labeled connected 3-chromatic (i.e., chromatic number = 3) graphs on n nodes.

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A084274 Number of labeled connected 4-chromatic (i.e., chromatic number = 4) graphs on n nodes.

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A228859 Triangular array read by rows. T(n,k) is the number of labeled bipartite graphs on n nodes having exactly k connected components; n>=1, 1<=k<=n.

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A228861 Irregular triangular array read by rows. T(n,k) is the number of connected labeled bipartite graphs on n nodes with exactly k edges; n >= 1, 0 <= k <= A002620(n+1).

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