A333160 -id:A333160 - cp's OEIS frontend

A333159 Triangle read by rows: T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column up to permutation of rows and columns.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 4, 12, 12, 4, 1, 1, 1, 1, 7, 31, 66, 31, 7, 1, 1, 1, 1, 8, 90, 433, 433, 90, 8, 1, 1, 1, 1, 12, 285, 3442, 7937, 3442, 285, 12, 1, 1, 1, 1, 14, 938, 30404, 171984, 171984, 30404, 938, 14, 1, 1

Offset: 0

Andrew Howroyd, Mar 10 2020

Rows and columns may be permuted independently. The case that rows and columns must be permuted together is covered by A333161.

T(n,k) is the number of k-regular bicolored graphs on 2n unlabeled nodes which are invariant when the two color classes are exchanged.

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 1,  1,   1;
  1, 1,  2,   1,    1;
  1, 1,  2,   2,    1,    1;
  1, 1,  4,   5,    4,    1,    1;
  1, 1,  4,  12,   12,    4,    1,   1;
  1, 1,  7,  31,   66,   31,    7,   1,  1;
  1, 1,  8,  90,  433,  433,   90,   8,  1, 1;
  1, 1, 12, 285, 3442, 7937, 3442, 285, 12, 1, 1;
  ...
The T(2,1) = 1 matrix is:
  [1 0]
  [0 1]
.
The T(4,2)= 2 matrices are:
  [1 1 0 0]   [1 1 0 0]
  [1 1 0 0]   [1 0 1 0]
  [0 0 1 1]   [0 1 0 1]
  [0 0 1 1]   [0 0 1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..230

Columns k=0..4 are A000012, A000012, A002865, A000840, A000843.
Row sums are A333160.
Central coefficients are A333165.
Cf. A008327, A122082, A333157, A133687, A333161.

T(n,k) = T(n,n-k).

A122082 Number of unlabeled bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged.

Original entry on oeis.org

1, 2, 5, 16, 67, 404, 3904, 64840, 1930842, 104698904, 10401039400, 1900637187280, 641429385018832, 401454435464761376, 467919402404052870944, 1019758699013228238271040, 4171161230867751509749228304

Offset: 0

N. J. A. Sloane, Oct 18 2006

nonn

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Andrew Howroyd, Table of n, a(n) for n = 0..50
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 of A123548.

Cf. A002724, A007139, A333160.

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]
a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]+1)\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 23 2017

a(n) = 2*A007139(n) - A002724(n). - Vladeta Jovovic, Feb 27 2007

More terms from Vladeta Jovovic, Feb 27 2007

A333681 Number of non-isomorphic n X n binary matrices with all row and column sums equal up to permutation of rows and columns.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 19, 44, 314, 7526, 846993, 324127860, 403254094632, 1555631972009430, 19731915624463099553, 791773335030637885025288, 107432353216118868234728540268, 47049030539260648478475949282317452, 71364337698829887974206671525372672234855

Offset: 0

Andrew Howroyd, Apr 01 2020

nonn

			The a(2) = 3 matrices are:
  [0 0]  [0 1]  [1 1]
  [0 0]  [1 0]  [1 1]

Row sums of A133687.

Cf. A000519, A067209, A333160.

a(n) = A000519(n) + 1.

A333732 Number of non-isomorphic n X n binary matrices with equal row and column sums up to permutation of rows and columns and transposition.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 18, 40, 230, 4296, 431206, 162267272, 201636689772, 777816803942188, 9865957936943931980, 395886667549681689592056, 53716176608076643470621240097, 23524515269630339982914646822137232, 35682168849414944013547274452501783251521

Offset: 0

Andrew Howroyd, Apr 03 2020

nonn

Number of simple regular bicolored graphs on 2n unlabeled nodes and allowing the color classes to be interchanged.

First differs from A008324 at n=12. See the note in A004066 by Sean A. Irvine for an explanation of why these two sequences are different.

Cf. A004066, A008324, A333160, A333681.

a(n) = (A333160(n) + A333681(n)) / 2.

a(n) >= A008324(n).

cp's OEIS Frontend

A333159 Triangle read by rows: T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column up to permutation of rows and columns.

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A122082 Number of unlabeled bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged.

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A333681 Number of non-isomorphic n X n binary matrices with all row and column sums equal up to permutation of rows and columns.

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A333732 Number of non-isomorphic n X n binary matrices with equal row and column sums up to permutation of rows and columns and transposition.

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