A283877 Number of non-isomorphic set-systems of weight n.

1, 1, 2, 4, 9, 18, 44, 98, 244, 605, 1595, 4273, 12048, 34790, 104480, 322954, 1031556, 3389413, 11464454, 39820812, 141962355, 518663683, 1940341269, 7424565391, 29033121685, 115921101414, 472219204088, 1961177127371, 8298334192288, 35751364047676, 156736154469354

Offset: 0

Author

Gus Wiseman, Mar 17 2017

Keywords

• nonn

Comments

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.

Examples

			Non-isomorphic representatives of the a(4)=9 set-systems are:
((1234)),
((1)(234)), ((3)(123)), ((12)(34)), ((13)(23)),
((1)(2)(12)), ((1)(2)(34)), ((1)(3)(23)),
((1)(2)(3)(4)).
		

Links

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

Crossrefs

Cf. A007716, A034691, A049311, A056156, A089259, A116540, A300913.

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, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + 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

Formula

Euler transform of A300913.

Extensions

a(0) = 1 prepended and terms a(11) and beyond from Andrew Howroyd, Sep 01 2019