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%I #15 Aug 27 2019 10:44:39
%S 1,2,10,208,45960,287800704,100103176111616,3837878984050795692032,
%T 32966965900633495618246298767360,
%U 128880214965936601447070466061615999984402432,464339910355487357558396669850788946402420533504952464572416
%N Number of non-isomorphic 3-uniform multiset systems over {1,...,n}.
%H Andrew Howroyd, <a href="/A320395/b320395.txt">Table of n, a(n) for n = 0..25</a>
%e Non-isomorphic representatives of the a(2) = 10 multiset systems:
%e {}
%e {{111}}
%e {{122}}
%e {{111}{222}}
%e {{112}{122}}
%e {{112}{222}}
%e {{122}{222}}
%e {{111}{122}{222}}
%e {{112}{122}{222}}
%e {{111}{112}{122}{222}}
%t Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Select[Tuples[Range[n],3],OrderedQ]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length],{prm,Permutations[Range[n]]}]/n!,{n,6}]
%o (PARI)
%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 rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L<k, listput(L, #L))); Vec(L)}
%o can(v, f)={my(d=1, u=v); while(d>0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
%o Q(perm)={my(t=0); forsubset([#perm+2, 3], v, t += can([v[1],v[2]-1,v[3]-2], t->perm[t])); t}
%o a(n)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(rep(p))); s/n!} \\ _Andrew Howroyd_, Aug 26 2019
%Y The 2-uniform case is A000666. The case of sets (as opposed to multisets) is A000665. The case of labeled spanning sets is A302374, with unlabeled case A322451.
%Y Cf. A000088, A000612, A003180, A070166, A301922, A317795, A319876.
%K nonn
%O 0,2
%A _Gus Wiseman_, Dec 12 2018
%E Terms a(9) and beyond from _Andrew Howroyd_, Aug 26 2019
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