A306005 Number of non-isomorphic set-systems of weight n with no singletons.

1, 0, 1, 1, 3, 4, 12, 19, 51, 106, 274, 647, 1773, 4664, 13418, 38861, 118690, 370588, 1202924, 4006557, 13764760, 48517672, 175603676, 651026060, 2471150365, 9590103580, 38023295735, 153871104726, 635078474978, 2671365285303, 11444367926725, 49903627379427

Offset

0,5

Comments

A set-system is a finite set of finite nonempty sets (edges). The weight is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Formula

a(n) = A283877(n) - A330053(n). - Gus Wiseman, Dec 09 2019

Example

Non-isomorphic representatives of the a(6) = 12 set-systems:
  {{1,2,3,4,5,6}}
  {{1,2},{3,4,5,6}}
  {{1,5},{2,3,4,5}}
  {{3,4},{1,2,3,4}}
  {{1,2,3},{4,5,6}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{3,4},{5,6}}
  {{1,2},{3,5},{4,5}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}

Prog

(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, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t), x, x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g, x, x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024

Crossrefs

The complement is counted by A330053.

Cf. A007716, A034691, A048143, A049311, A054921, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

Sequence in context: A347257 A123765 A124817 * A124818 A321412 A124638

Adjacent sequences: A306002 A306003 A306004 * A306006 A306007 A306008

Keyword

nonn

Author

Gus Wiseman, Jun 16 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

Status

approved