A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

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

Offset: 1

Author

Peter J. Cameron

Keywords

• nonn
• nice

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

Examples

			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
		

Links

• Aliaksandr Siarhei, Table of n, a(n) for n = 1..102
• Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Peter J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
• Peter J. Cameron, Problems on Permutation Groups, see Problem 3
• Index entries for sequences related to binary matrices.

Crossrefs

Main diagonal of A321609.

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

Cf. A057149, A057150, A057151, A057152.

Cf. A034691, A056156, A089259, A116540, A283877.

Programs

PARI

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 16 2023

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.

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