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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

References

• 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).

Links

• 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

Crossrefs

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.

Programs

Maple

# See Marko Riedel link.

Mathematica

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 *)

PARI

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

Formula

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

Extensions

More terms from Vladeta Jovovic, Feb 04 2000

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