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A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.
(Formerly M1801 N0711)
38
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
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]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
M. Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
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
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 *)
PROG
(PARI) a(n) = A(n, n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023
CROSSREFS
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
KEYWORD
nonn,nice
AUTHOR
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 June 14 21:35 EDT 2024. Contains 373401 sequences. (Running on oeis4.)