1,2

Also the number of binary matrices with n ones, with no zero rows or columns, up to row and column permutation. Also the number of bipartite graphs with n edges, no isolated vertices, and a distinguished bipartite block, up to isomorphism.

The EULERi transform is also interesting.

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.

Solution of problem 3 on Cameron's list of permutation group problems

Index entries for sequences related to binary matrices

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, then apply EULER transform.

a(n) is coefficient of x^n in cycle index Z(S_n X S_n; x_1, x_2, ...) if we replace x_i by 1+x^i, where S_n X S_n is Cartesian product of symmetric groups S_n of degree n.

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]

Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724.

Sequence in context: A052370 A053304 A053305 this_sequence A068590 A130095 A072824

Adjacent sequences: A049308 A049309 A049310 this_sequence A049312 A049313 A049314

nonn,nice,more

Peter Cameron (p.j.cameron(AT)qmw.ac.uk)

More terms and formula from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 29 2000