A317775 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.

1, 3, 10, 36, 136, 596, 2656, 13187, 68226, 381572, 2233091, 13940407, 90981030, 626911429, 4509031955, 33987610040, 266668955183, 2180991690286, 18512572760155, 163103174973092, 1487228204311039, 14027782824491946, 136585814043190619, 1371822048393658001, 14190528438090988629

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..50

Example

The a(3) = 10 strict multiset partitions:
  {{1,1,1}}, {{1},{1,1}},
  {{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
  {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n], UnsameQ@@#&]], {n, 6}]

Prog

(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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]!)}
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

Crossrefs

Cf. A001055, A007716, A045778, A281116, A317449, A317584, A317654, A317755, A317757, A317776.

Sequence in context: A026854 A136576 A129156 * A171753 A002212 A340941

Adjacent sequences: A317772 A317773 A317774 * A317776 A317777 A317778

Keyword

nonn

Author

Gus Wiseman, Aug 06 2018

Extensions

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

Status

approved