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%I #10 Jan 27 2024 14:13:07
%S 1,0,0,0,1,0,4,4,16,26,87,181,570,1453,4464,13038,41548,132217,442603,
%T 1506803,5305174,19092816,70548770,266495254,1029835424,4063610148,
%U 16366919221,67217627966,281326631801,1199048810660,5201341196693,22950740113039,102957953031700
%N Number of non-isomorphic set-systems of weight n with no endpoints.
%C A set-system is a finite set of finite nonempty set of positive integers. 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.
%H Andrew Howroyd, <a href="/A330054/b330054.txt">Table of n, a(n) for n = 0..50</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Degree_(graph_theory)">Degree (graph theory)</a>
%e Non-isomorphic representatives of the a(0) = 1 through a(8) = 16 multiset partitions (empty columns not shown):
%e 0 {1}{2}{12} {12}{13}{23} {13}{23}{123} {12}{134}{234}
%e {1}{23}{123} {1}{3}{23}{123} {1}{234}{1234}
%e {1}{2}{13}{23} {3}{12}{13}{23} {12}{34}{1234}
%e {1}{2}{3}{123} {1}{2}{3}{13}{23} {1}{12}{34}{234}
%e {12}{13}{24}{34}
%e {1}{2}{134}{234}
%e {1}{2}{34}{1234}
%e {2}{13}{14}{234}
%e {2}{13}{23}{123}
%e {3}{13}{23}{123}
%e {1}{2}{13}{24}{34}
%e {1}{2}{3}{14}{234}
%e {1}{2}{3}{23}{123}
%e {1}{2}{3}{4}{1234}
%e {2}{3}{12}{13}{23}
%e {1}{2}{3}{4}{12}{34}
%o (PARI)
%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o 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}
%o K(q, t, k)={my(g=1+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)}
%o 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)/t,x,x^t) )), n)); s/n!)} \\ _Andrew Howroyd_, Jan 27 2024
%Y The complement is counted by A330052.
%Y The multiset partition version is A302545.
%Y Non-isomorphic set-systems with no singletons are A306005.
%Y Non-isomorphic set-systems counted by vertices are A000612.
%Y Non-isomorphic set-systems counted by weight are A283877.
%Y Cf. A007716, A055621, A317533, A317794, A319559, A320665, A330055, A330056, A330058.
%K nonn
%O 0,7
%A _Gus Wiseman_, Nov 30 2019
%E a(11) onwards from _Andrew Howroyd_, Jan 27 2024
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