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%I #9 Dec 30 2020 17:33:27
%S 1,3,10,36,136,596,2656,13187,68226,381572,2233091,13940407,90981030,
%T 626911429,4509031955,33987610040,266668955183,2180991690286,
%U 18512572760155,163103174973092,1487228204311039,14027782824491946,136585814043190619,1371822048393658001,14190528438090988629
%N Number of strict multiset partitions of strongly normal multisets of size n, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.
%H Andrew Howroyd, <a href="/A317775/b317775.txt">Table of n, a(n) for n = 1..50</a>
%e The a(3) = 10 strict multiset partitions:
%e {{1,1,1}}, {{1},{1,1}},
%e {{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
%e {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.
%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],UnsameQ@@#&]],{n,6}]
%o (PARI)
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1,-n)/prod(i=1, #v, i^v[i]*v[i]!)}
%o seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p,n))); s[n]+=1; s/2} \\ _Andrew Howroyd_, Dec 30 2020
%Y Cf. A001055, A007716, A045778, A281116, A317449, A317584, A317654, A317755, A317757, A317776.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 06 2018
%E Terms a(10) and beyond from _Andrew Howroyd_, Dec 30 2020
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