A330783 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.

1, 1, 3, 8, 27, 94, 385, 1673, 8079, 41614, 231447, 1364697, 8559575, 56544465, 393485452, 2867908008, 21869757215, 173848026202, 1438593095272, 12360614782433, 110119783919367, 1015289796603359, 9674959683612989, 95147388659652754, 964559157655032720, 10067421615492769230

Offset

0,3

Comments

The (weakly) normal version is A116540.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Example

The a(1) = 1 through a(3) = 8 set multipartitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
The a(4) = 27 set multipartitions:
  {{1},{1},{1},{1}}  {{1},{1},{1,2}}  {{1},{1,2,3}}  {{1,2,3,4}}
  {{1},{1},{1},{2}}  {{1},{1},{2,3}}  {{1,2},{1,2}}
  {{1},{1},{2},{2}}  {{1},{2},{1,2}}  {{1,2},{1,3}}
  {{1},{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{2,3,4}}
  {{1},{2},{3},{4}}  {{1},{2},{3,4}}  {{1,2},{3,4}}
                     {{1},{3},{1,2}}  {{1,3},{2,4}}
                     {{1},{3},{2,4}}  {{1,4},{2,3}}
                     {{1},{4},{2,3}}  {{2},{1,3,4}}
                     {{2},{3},{1,4}}  {{3},{1,2,4}}
                     {{2},{4},{1,3}}  {{4},{1,2,3}}
                     {{3},{4},{1,2}}

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], And@@UnsameQ@@@#&]], {n, 0, 5}]

Prog

(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p, n)); s} \\ Andrew Howroyd, Dec 30 2020

Crossrefs

Allowing edges to be multisets gives is A035310.

The strict case is A318402.

The constant case is A000005.

The (weakly) normal version is A116540.

Unlabeled set multipartitions are A049311.

Set multipartitions of prime indices are A050320.

Set multipartitions of integer partitions are A089259.

Cf. A001055, A047968, A255906, A269134, A283877, A296119, A317775, A318360, A318362, A330625, A330628.

Sequence in context: A259811 A148841 A148842 * A148843 A340334 A319787

Adjacent sequences: A330780 A330781 A330782 * A330784 A330785 A330786

Keyword

nonn

Author

Gus Wiseman, Jan 02 2020

Extensions

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

Status

approved