A363846 -id:A363846 - cp's OEIS frontend

A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.

1, 2, 7, 36, 317, 5624, 251610, 33642660, 14685630688, 21467043671008, 105735224248507784, 1764356230257807614296, 100455994644460412263071692, 19674097197480928600253198363072, 13363679231028322645152300040033513414, 31735555932041230032311939400670284689732948

Offset: 0

N. J. A. Sloane

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Also, number of bipartite graphs with both partite sets of size n, one of which is marked. For connected bipartite graphs, see A363846. - Max Alekseyev, Jun 24 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).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..26 from Alois P. Heinz)
Manuel Kauers and Jakob Moosbauer, Good pivots for small sparse matrices, arXiv:2006.01623 [cs.SC], 2020.
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
Mathematics Stack Exchange, How many n-by-m binary matrices are there up to row and column permutations
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Marko Riedel, Maple code with two different algorithms
M. Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
Index entries for sequences related to binary matrices
Index to number of inequivalent matrices modulo permutation of rows and columns

Cf. A002623, A002727, A006148, A002728, A002725, A052269, A052271, A052272, A091059, A363845, A363846, A000595.
Cf. A028657 (this sequence is the diagonal). - N. J. A. Sloane, Sep 01 2013
Column k=2 of A246106.

Maple
```
# See Marko Riedel link.
```

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Union[Flatten[Table[ 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]];
Table[A[n, n], {n, 0, 15}] (* Jean-François Alcover, Aug 10 2018, after Alois P. Heinz *)

a(n) = A(n,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n} (fixA[s_1, s_2, ...;t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fixA[...] = 2^Sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Christian G. Bower, Dec 18 2003

a(n) = A028657(2*n, n). - Max Alekseyev, Jun 24 2023

More terms from Vladeta Jovovic, Feb 04 2000

a(15) from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

A363845 Triangle read by rows: T(n,k) = number of connected n-node graphs with k nodes in distinguished bipartite block, k = 0..n.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 6, 13, 6, 1, 0, 0, 1, 9, 34, 34, 9, 1, 0, 0, 1, 12, 76, 150, 76, 12, 1, 0, 0, 1, 16, 155, 558, 558, 155, 16, 1, 0, 0, 1, 20, 290, 1824, 3529, 1824, 290, 20, 1, 0, 0, 1, 25, 510, 5375, 19687, 19687, 5375, 510, 25, 1, 0, 0, 1, 30, 853, 14549, 98726, 194203, 98726, 14549, 853, 30, 1, 0

Offset: 0

Max Alekseyev, Jun 24 2023

Also, rectangular array read by antidiagonals: A(m,n) = number of distinct m X n binary matrices M up to permutations of rows or columns such that M represents the reduced adjacency matrix of a connected bipartite graph.

			n=0: 1,
n=1: 1, 1,
n=2: 0, 1, 0,
n=3: 0, 1, 1, 0,
n=4: 0, 1, 2, 1, 0,
n=5: 0, 1, 4, 4, 1, 0,
n=6: 0, 1, 6, 13, 6, 1, 0,
n=7: 0, 1, 9, 34, 34, 9, 1, 0,
n=8: 0, 1, 12, 76, 150, 76, 12, 1, 0,
n=9: 0, 1, 16, 155, 558, 558, 155, 16, 1, 0,
...

Max Alekseyev, Rows n = 0..45, flattened

Inverse bivariate Euler transform of A028657.

Cf. A002724, A363846.

cp's OEIS Frontend

A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.

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A363845 Triangle read by rows: T(n,k) = number of connected n-node graphs with k nodes in distinguished bipartite block, k = 0..n.

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A363884 Number of connected bipartite graphs on 2n nodes with a distinguished partite set of size n and with a unimodular reduced adjacency matrix.

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