

A049311


Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.


116



1, 3, 6, 16, 34, 90, 211, 558, 1430, 3908, 10725, 30825, 90156, 273234, 848355, 2714399, 8909057, 30042866, 103859678, 368075596, 1335537312, 4958599228, 18820993913, 72980867400, 288885080660, 1166541823566, 4802259167367, 20141650236664
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OFFSET

1,2


COMMENTS

Also the number of bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism.
The EULERi transform (A056156) is also interesting.
a(n) is also the number of nonisomorphic set multipartitions (multisets of sets) of weight n.  Gus Wiseman, Mar 17 2017


LINKS

Aliaksandr Siarhei, Table of n, a(n) for n = 1..102
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
P. J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386394.
P. J. Cameron, Solution of problem 3 on Cameron's list of permutation group problems
Index entries for sequences related to binary matrices


FORMULA

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, then apply EULER transform.
a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; 1+x, 1+x^2, ...), where S_n X S_n is Cartesian product of symmetric groups S_n of degree n.


EXAMPLE

E.g. a(2) = 3: two ones in same row, two ones in same column, or neither.
a(3) = 6 is coefficient of x^3 in (1/36)*((1 + x)^9 + 6*(1 + x)^3*(1 + x^2)^3 + 8*(1 + x^3)^3 + 9*(1 + x)*(1 + x^2)^4 + 12*(1 + x^3)*(1 + x^6))=1 + x + 3*x^2 + 6*x^3 + 7*x^4 + 7*x^5 + 6*x^6 + 3*x^7 + x^8 + x^9.
There are a(3) = 6 binary matrices with 3 ones, with no zero rows or columns, up to row and column permutation:
[1 0 0] [1 1 0] [1 0] [1 1] [1 1 1] [1]
[0 1 0] [0 0 1] [1 0] [1 0] ....... [1].
[0 0 1] ....... [0 1] ............. [1]
Nonisomorphic representatives of the a(3)=6 set multipartitions are: ((123)), ((1)(23)), ((2)(12)), ((1)(1)(1)), ((1)(2)(2)), ((1)(2)(3)).  Gus Wiseman, Mar 17 2017


CROSSREFS

Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724.
Cf. A057149, A057150, A057151, A057152.
Cf. A034691, A056156, A089259, A116540, A283877.
Sequence in context: A052370 A053304 A053305 * A068590 A319752 A130095
Adjacent sequences: A049308 A049309 A049310 * A049312 A049313 A049314


KEYWORD

nonn,nice


AUTHOR

Peter J. Cameron


EXTENSIONS

More terms and formula from Vladeta Jovovic, Jul 29 2000
a(19)a(28) from Max Alekseyev, Jul 22 2009
a(29)a(102) from Aliaksandr Siarhei, Dec 13 2013
Name edited by Gus Wiseman, Dec 18 2018


STATUS

approved



