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%I #11 Dec 30 2020 14:58:46
%S 1,1,3,8,27,94,385,1673,8079,41614,231447,1364697,8559575,56544465,
%T 393485452,2867908008,21869757215,173848026202,1438593095272,
%U 12360614782433,110119783919367,1015289796603359,9674959683612989,95147388659652754,964559157655032720,10067421615492769230
%N Number of set multipartitions (multisets of sets) of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
%C The (weakly) normal version is A116540.
%H Andrew Howroyd, <a href="/A330783/b330783.txt">Table of n, a(n) for n = 0..50</a>
%e The a(1) = 1 through a(3) = 8 set multipartitions:
%e {{1}} {{1,2}} {{1,2,3}}
%e {{1},{1}} {{1},{1,2}}
%e {{1},{2}} {{1},{2,3}}
%e {{2},{1,3}}
%e {{3},{1,2}}
%e {{1},{1},{1}}
%e {{1},{1},{2}}
%e {{1},{2},{3}}
%e The a(4) = 27 set multipartitions:
%e {{1},{1},{1},{1}} {{1},{1},{1,2}} {{1},{1,2,3}} {{1,2,3,4}}
%e {{1},{1},{1},{2}} {{1},{1},{2,3}} {{1,2},{1,2}}
%e {{1},{1},{2},{2}} {{1},{2},{1,2}} {{1,2},{1,3}}
%e {{1},{1},{2},{3}} {{1},{2},{1,3}} {{1},{2,3,4}}
%e {{1},{2},{3},{4}} {{1},{2},{3,4}} {{1,2},{3,4}}
%e {{1},{3},{1,2}} {{1,3},{2,4}}
%e {{1},{3},{2,4}} {{1,4},{2,3}}
%e {{1},{4},{2,3}} {{2},{1,3,4}}
%e {{2},{3},{1,4}} {{3},{1,2,4}}
%e {{2},{4},{1,3}} {{4},{1,2,3}}
%e {{3},{4},{1,2}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t Table[Length[Select[Join@@mps/@strnorm[n],And@@UnsameQ@@@#&]],{n,0,5}]
%o (PARI)
%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n)))/prod(i=1, #v, i^v[i]*v[i]!)}
%o seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ _Andrew Howroyd_, Dec 30 2020
%Y Allowing edges to be multisets gives is A035310.
%Y The strict case is A318402.
%Y The constant case is A000005.
%Y The (weakly) normal version is A116540.
%Y Unlabeled set multipartitions are A049311.
%Y Set multipartitions of prime indices are A050320.
%Y Set multipartitions of integer partitions are A089259.
%Y Cf. A001055, A047968, A255906, A269134, A283877, A296119, A317775, A318360, A318362, A330625, A330628.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 02 2020
%E Terms a(10) and beyond from _Andrew Howroyd_, Dec 30 2020
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