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A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.
(Formerly M1801 N0711)
35
1, 2, 7, 36, 317, 5624, 251610, 33642660, 14685630688, 21467043671008, 105735224248507784, 1764356230257807614296, 100455994644460412263071692, 19674097197480928600253198363072, 13363679231028322645152300040033513414, 31735555932041230032311939400670284689732948 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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

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 *)

CROSSREFS

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

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Feb 04 2000

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

STATUS

approved

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Last modified February 8 02:43 EST 2023. Contains 360133 sequences. (Running on oeis4.)