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A319751
Number of non-isomorphic set systems of weight n with empty intersection.
3
1, 0, 1, 2, 6, 13, 35, 83, 217, 556, 1504, 4103, 11715, 34137, 103155, 320217, 1025757, 3376889, 11436712, 39758152, 141817521, 518322115, 1939518461, 7422543892, 29028055198, 115908161428, 472185530376, 1961087909565, 8298093611774, 35750704171225, 156734314212418
OFFSET
0,4
COMMENTS
A set system is a finite set of finite nonempty sets. Its weight is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
LINKS
EXAMPLE
Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 set systems:
2: {{1},{2}}
3: {{1},{2,3}}
{{1},{2},{3}}
4: {{1},{2,3,4}}
{{1,2},{3,4}}
{{1},{2},{1,2}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{1},{2},{3},{4}}
5: {{1},{2,3,4,5}}
{{1,2},{3,4,5}}
{{1},{2},{3,4,5}}
{{1},{4},{2,3,4}}
{{1},{2,3},{4,5}}
{{1},{2,4},{3,4}}
{{2},{3},{1,2,3}}
{{2},{1,3},{2,3}}
{{4},{1,2},{3,4}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{4,5}}
{{1},{2},{4},{3,4}}
{{1},{2},{3},{4},{5}}
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, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q, n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t]-subst(u[t], x, x^2), O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t] - subst(x^t*u[t], x, x^2), O(x*x^n)))*(1+x), n)); s/n!)} \\ Andrew Howroyd, May 30 2023
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 27 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, May 30 2023
STATUS
approved

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Last modified September 21 15:43 EDT 2024. Contains 376087 sequences. (Running on oeis4.)