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A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations. 125
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 (list; graph; refs; listen; history; text; internal format)
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 non-isomorphic 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), 386-394.

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]

Non-isomorphic 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 A327736 A319752

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

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)