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A330055
Number of non-isomorphic set-systems of weight n with no singletons or endpoints.
11
1, 0, 0, 0, 0, 0, 1, 1, 3, 5, 16, 24, 90, 179, 567, 1475, 4623, 13650, 44475, 144110, 492017, 1706956, 6124330, 22442687, 84406276, 324298231, 1273955153, 5106977701, 20885538133, 87046940269, 369534837538, 1596793560371, 7019424870960, 31374394197536, 142514998263015
OFFSET
0,9
COMMENTS
A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system 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(7) = 1 through a(10) = 16 set-systems:
{12}{13}{123} {12}{134}{234} {12}{134}{1234} {12}{1345}{2345}
{12}{34}{1234} {123}{124}{134} {123}{124}{1234}
{12}{13}{24}{34} {12}{13}{14}{234} {123}{145}{2345}
{12}{13}{23}{123} {12}{345}{12345}
{12}{13}{24}{134} {12}{13}{124}{134}
{12}{13}{124}{234}
{12}{13}{14}{1234}
{12}{13}{24}{1234}
{12}{13}{245}{345}
{12}{13}{45}{2345}
{12}{34}{123}{124}
{12}{34}{125}{345}
{12}{34}{135}{245}
{13}{24}{123}{124}
{12}{13}{14}{23}{24}
{12}{13}{24}{35}{45}
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)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g-subst(g, x, x^2)}
S(q, t, k)={(x-x^2)*sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(K(q, t, n\t)-S(q, t, n\t), x, x^t)/t )), n)); s/n!)} \\ Andrew Howroyd, Jan 27 2024
CROSSREFS
The labeled version is A330056.
The "multi" version is A320665.
Non-isomorphic set-systems with no singletons are A306005.
Non-isomorphic set-systems with no endpoints are A330054.
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Sequence in context: A039782 A331638 A080056 * A019096 A370604 A295358
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 30 2019
EXTENSIONS
a(11) onwards from Andrew Howroyd, Jan 27 2024
STATUS
approved