A002027 -id:A002027 - cp's OEIS frontend

A002031 Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.

2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822

Offset: 2

N. J. A. Sloane

Also number of labeled connected graphs with 2-colored nodes with no isolated nodes where black nodes are only connected to white nodes and vice versa.

In- or outdegree zero implies loops are not admitted. Multi-arcs are not admitted. - R. J. Mathar, Nov 18 2023

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).

Alois P. Heinz, Table of n, a(n) for n = 2..100
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.
N. J. A. Sloane, Transforms

Cf. A001831, A001832, A002032, A047863, A052332, A007776 (unlabeled case). Essentially the same as A002027.

logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: digr:= n-> add(binomial(n,k) *(2^k-2)^(n-k), k=0..n): a:= logtr(digr): seq(a(n), n=2..25);  # Alois P. Heinz, Sep 14 2008

terms = 17; s = Log[Sum[Exp[(2^n - 2)*x]*(x^n/n!), {n, 0, terms+2}]] + O[x]^(terms+2); Drop[CoefficientList[s, x]*Range[0, terms+1]!, 2] (* Jean-François Alcover, Nov 08 2011, after Vladeta Jovovic, updated Jan 12 2018 *)

Logarithmic transform of A052332.
E.g.f.: log(Sum(exp((2^n-2)*x)*x^n/n!, n=0..infinity)). - Vladeta Jovovic, May 28 2004
a(n) = f(n,2) using functions defined in A002032. - Sean A. Irvine, May 29 2013

More terms, formula and new title from Christian G. Bower, Dec 15 1999

Corrected by Vladeta Jovovic, Apr 12 2003

A322279 Array read by antidiagonals: T(n,k) is the number of connected graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 6, 6, 0, 0, 1, 5, 12, 42, 38, 0, 0, 1, 6, 20, 132, 618, 390, 0, 0, 1, 7, 30, 300, 3156, 15990, 6062, 0, 0, 1, 8, 42, 570, 9980, 136980, 668526, 134526, 0, 0, 1, 9, 56, 966, 24330, 616260, 10015092, 43558242, 4172198, 0, 0

Offset: 0

Andrew Howroyd, Dec 01 2018

Not all colors need to be used.

			Array begins:
===============================================================
n\k| 0 1      2        3          4           5           6
---+-----------------------------------------------------------
0  | 1 1      1        1          1           1           1 ...
1  | 0 1      2        3          4           5           6 ...
2  | 0 0      2        6         12          20          30 ...
3  | 0 0      6       42        132         300         570 ...
4  | 0 0     38      618       3156        9980       24330 ...
5  | 0 0    390    15990     136980      616260     1956810 ...
6  | 0 0   6062   668526   10015092    65814020   277164210 ...
7  | 0 0 134526 43558242 1199364852 11878194300 67774951650 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Columns k=2..5 are A002027, A002028, A002029, A002030.

Cf. A058843, A058875, A322278, A322280.

M(n)={
  my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
  my(q=sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n));
  my(W=Mat(vector(n, k, Col(serlaplace(1 + log(serconvol(q, p^k)))))));
  matconcat([1, W]);
}
my(T=M(7)); for(n=1, #T, print(T[n,]))

k-th column is the logarithmic transform of the k-th column of A322280.

E.g.f of k-th column: 1 + log(Sum_{n>=0} A322280(n,k)*x^n/n!).

A002032 Number of n-colored connected graphs on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 24, 912, 87360, 19226880, 9405930240, 10142439229440, 24057598104207360, 125180857812868300800, 1422700916050060841779200, 35136968950395142864227532800, 1876028272361273394915958613606400, 215474119792145796020405035320528076800

Offset: 0

N. J. A. Sloane

nonn

Every connected graph on n nodes can be colored with n colors in exactly n! ways, so this sequence is just n! * A001187(n). - Andrew Howroyd, Dec 03 2018

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).

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Cf. A002027. A002028, A002029, A002030, A002031.

Cf. A001187, A322278, A322279, A322280.

(* b = A001187 *) b[n_] := b[n] = If[n == 0, 1, 2^(n(n-1)/2) - Sum[k* Binomial[n, k]*2^((n-k)(n-k-1)/2)*b[k], {k, 1, n-1}]/n];
a[n_] := n! b[n];
Array[a, 14] (* Jean-François Alcover, Aug 16 2019, using Alois P. Heinz's code for A001187 *)

seq(n) = {Vec(serlaplace(serlaplace(1 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n))))))} \\ Andrew Howroyd, Dec 03 2018

a(n) = n!*A001187(n). - Andrew Howroyd, Dec 03 2018
Define M_0(k)=1, M_n(0)=0, M_n(k) = Sum_{r=0..n} C(n,r)*2^(r*(n-r))*M_r(k-1) [M_n(k) = A322280(n,k)], m_n(k) = M_n(k) -Sum_{r=1..n-1} C(n-1,r-1)*m_r(k)*M_{n-r}(k) [m_n(k) = A322279(n,k)], f_n(k) = Sum_{r=1..k} (-1)^(k-r)*C(k,r)*m_n(r). This sequence gives a(n) = f_n(n). - Sean A. Irvine, May 29 2013, edited Andrew Howroyd, Dec 03 2018
The above formula is referenced by sequences A002027-A002030, A002031. - Andrew Howroyd, Dec 03 2018

More terms from Sean A. Irvine, May 29 2013
Name clarified by Andrew Howroyd, Dec 03 2018
a(0)=1 prepended by Andrew Howroyd, Jan 05 2024

A342501 T(n,k) is the number of connected labeled posets with n elements and rank k: triangle read by rows.

Original entry on oeis.org

1, 0, 2, 0, 6, 6, 0, 38, 84, 24, 0, 390, 1710, 840, 120, 0, 6062, 49740, 36840, 8280, 720, 0, 134526, 2050566, 2184000, 646800, 85680, 5040, 0, 4172198, 118645044, 177549624, 65313360, 10735200, 947520, 40320

Offset: 1

R. J. Mathar, Mar 14 2021

This is a variant of A342587 admitting only connected posets.

			The table starts in row n=1 and shows ranks k>=0:
1: 1
2: 0 2
3: 0 6 6
4: 0 38 84 24
5: 0 390 1710 840 120
6: 0 6062 49740 36840 8280 720
7: 0 134526 2050566 2184000 646800 85680 5040
8: 0 4172198 118645044 177549624 65313360 10735200 947520 40320

Index entries for sequences related to posets

Cf. A001927 (row sums), A000142 (diagonal), A002031/A002027 (rank 1), A342500 (unlabeled).

cp's OEIS Frontend

A002031 Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.

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A322279 Array read by antidiagonals: T(n,k) is the number of connected graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.

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A002032 Number of n-colored connected graphs on n labeled nodes.

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A342501 T(n,k) is the number of connected labeled posets with n elements and rank k: triangle read by rows.

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