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A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.
(Formerly M1801 N0711)
1, 2, 7, 36, 317, 5624, 251610, 33642660, 14685630688, 21467043671008, 105735224248507784, 1764356230257807614296, 100455994644460412263071692, 19674097197480928600253198363072, 13363679231028322645152300040033513414, 31735555932041230032311939400670284689732948 (list; graph; refs; listen; history; text; internal format)



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


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, 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]

Math StackExchange, 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


a(n) = sum {1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n} (fix A[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 fix A[...] = 2^sum {i, j>=1} (gcd(i, j)*s_i*t_j). - Christian G. Bower, Dec 18 2003


# 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 *)


Cf. A002623, A002727, A006148, A002728, A002725, A052269, A052271, A052272, A091059.

Cf. A028657 (this sequence is the diagonal). - N. J. A. Sloane, Sep 01 2013

Column k=2 of A246106.

Sequence in context: A012717 A072236 A007474 * A348106 A292206 A203900

Adjacent sequences:  A002721 A002722 A002723 * A002725 A002726 A002727




N. J. A. Sloane


More terms from Vladeta Jovovic, Feb 04 2000

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

Added Maple code to compute the cycle indices of the group permuting the slots of the matrix and hence the values of the sequence, Marko Riedel, Jul 27 2014

Added Maple implementation of the closed form result by C. G. Bower and of the textbook method, Marko Riedel, Jul 28 2014

Moved Maple code from inline to attachment by Marko Riedel, Dec 27 2014



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Last modified November 30 13:47 EST 2021. Contains 349420 sequences. (Running on oeis4.)