A007140 -id:A007140 - cp's OEIS frontend

A122081 Duplicate of A007140.

1, 1, 3, 14, 115, 2086, 101791, 14835870, 6852422567, 10338780759514, 51804974736769271

Offset: 0

dead

A007139 Number of unlabeled bicolored bipartite graphs on 2n nodes having n nodes of each color with no edges between vertices of the same color and allowing the color classes to be interchanged.

1, 2, 6, 26, 192, 3014, 127757, 16853750, 7343780765, 10733574184956, 52867617324773592, 882178116079222400788, 50227997322550920824045262, 9837048598740665027344331562224, 6681839615514161556535851222043192179, 15867777966020615016665849049841756464001994

Offset: 0

N. J. A. Sloane

nonn

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Marko Riedel, Table of n, a(n) for n = 0..30
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy]
Marko Riedel, Cycle indices for K_{n,n} including shuffles and reflections.
Marko Riedel, Maple code for cycle indices of K_{n,n}

Cf. A007140.

a(n) = (A002724(n)+A122082(n))/2. - Vladeta Jovovic, Feb 27 2007

More terms from Vladeta Jovovic, Feb 27 2007
Edited by N. J. A. Sloane, Aug 14 2007
Name clarified by Marko Riedel, Jun 15 2017
More terms from Marko Riedel, Jun 16 2017

A123547 Triangle read by rows: T(n,k) = number of unlabeled bicolored graphs with no isolated nodes having 2n nodes and k edges, with n nodes of each color. Here n >= 0, 0 <= k <= n^2.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 4, 3, 2, 1, 1, 0, 0, 0, 0, 1, 2, 8, 14, 21, 20, 20, 12, 9, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1, 2, 10, 31, 76, 137, 221, 285, 321, 301, 253, 182, 122, 69, 38, 19, 10, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 11, 43, 162, 451, 1121, 2314, 4255, 6702

Offset: 0

N. J. A. Sloane, Nov 14 2006

The colors may be interchanged.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

R. W. Robinson, Rows 0 through 7, flattened
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy]

Row sums give A007140. Cf. A007139, A106498 (gives beginning of this triangle).

A106498 Triangle read by rows: T(n,k) = number of unlabeled bicolored graphs with isolated nodes allowed having 2n nodes and k edges, with n nodes of each color. Here n >= 0, 0 <= k <= n^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 4, 5, 5, 4, 2, 1, 1, 1, 1, 2, 4, 10, 13, 23, 26, 32, 26, 23, 13, 10, 4, 2, 1, 1, 1, 1, 2, 4, 10, 20, 39, 72, 128, 198, 280, 353, 399, 399, 353, 280, 198, 128, 72, 39, 20, 10, 4, 2, 1, 1, 1, 1, 2, 4, 10, 20, 50, 99, 227, 458, 934, 1711

Offset: 0

N. J. A. Sloane, Nov 14 2006

The colors may be interchanged.

			Triangles A106498 and A123547 begin:
n = 0
k = 0 : 1, 1
Total = 1, 1
n = 1
k = 0 : 1, 0
k = 1 : 1, 1
Total = 2, 1
n = 2
k = 0 : 1, 0
k = 1 : 1, 0
k = 2 : 2, 1
k = 3 : 1, 1
k = 4 : 1, 1
Totals = 6, 3
n = 3
k = 0 : 1, 0
k = 1 : 1, 0
k = 2 : 2, 0
k = 3 : 4, 1
k = 4 : 5, 2
k = 5 : 5, 4
k = 6 : 4, 3
k = 7 : 2, 2
k = 8 : 1, 1
k = 9 : 1, 1
Totals = 26, 14

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

R. W. Robinson, Rows 0 through 7, flattened
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy]

Row sums give A007139. Cf. A007140, A123547.

A319155 Number of bicolored graphs on 2n unlabeled nodes without isolated nodes and which are invariant when the two color classes are interchanged.

Original entry on oeis.org

1, 1, 3, 11, 51, 337, 3500, 60936, 1866002, 102768062, 10296340496, 1890236147880, 639528747831552, 400813006079742544, 467517947968588109568, 1019290779610824185400096, 4170141472168738281510957264, 32130367702064742239376997422512

Offset: 0

Andrew Howroyd, Sep 25 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007140, A122082.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v + 1, 2];
A122082[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
a[n_] := A122082[n] - A122082[n-1];
a /@ Range[0, 17] (* Jean-François Alcover, Sep 05 2019, after Andrew Howroyd in A122082 *)

a(n) = A122082(n) - A122082(n-1).

cp's OEIS Frontend

A122081 Duplicate of A007140.

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A007139 Number of unlabeled bicolored bipartite graphs on 2n nodes having n nodes of each color with no edges between vertices of the same color and allowing the color classes to be interchanged.

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A123547 Triangle read by rows: T(n,k) = number of unlabeled bicolored graphs with no isolated nodes having 2n nodes and k edges, with n nodes of each color. Here n >= 0, 0 <= k <= n^2.

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A106498 Triangle read by rows: T(n,k) = number of unlabeled bicolored graphs with isolated nodes allowed having 2n nodes and k edges, with n nodes of each color. Here n >= 0, 0 <= k <= n^2.

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A319155 Number of bicolored graphs on 2n unlabeled nodes without isolated nodes and which are invariant when the two color classes are interchanged.

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