A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

1, 1, 0, 2, 1, 3, 0, 7, 3, 11, 18, 9

Offset

0,4

Comments

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

Links

Table of n, a(n) for n=0..11.

Example

The normal multiset {1,2,2,2,2,3,3,4} has three multiset partitions into a set of sets:
  {{2},{1,2},{2,3},{2,3,4}}
  {{2},{2,3},{2,4},{1,2,3}}
  {{2},{3},{1,2},{2,3},{2,4}}
so is not counted under a(8).
The a(1) = 1 through a(7) = 7 normal multisets:
  {1}  .  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  .  {1,1,1,1,2,3,4}
          {1,2,2}             {1,2,2,2,3}     {1,1,1,2,2,2,3}
                              {1,2,3,3,3}     {1,1,1,2,3,3,3}
                                              {1,2,2,2,2,3,4}
                                              {1,2,2,2,3,3,3}
                                              {1,2,3,3,3,3,4}
                                              {1,2,3,4,4,4,4}

Mathematica

allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]] /@ Subsets[Range[n-1]+1]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]] /@ Cases[Subsets[set], {i, ___}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n], Length[Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]]==1&]], {n, 0, 5}]

Crossrefs

For constant instead of strict blocks we have A000045.

Factorizations of this type are counted by A050326, with distinct sums A381633.

For the strong case see A292444, A382430, complement A381996, A382523.

MM-numbers of sets of sets are A302494, see A302478, A382201.

Twice-partitions into distinct sets are counted by A358914, with distinct sums A279785.

For integer partitions we have A382079 (A293511), with distinct sums A382460, (A381870).

With distinct sums we have A382459.

Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360.

Normal multiset partitions: A034691, A035310, A116539, A255906, A381718.

Set systems: A050342, A296120, A318361.

Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).

Cf. A000110, A000670, A007716, A255903, A275780, A292432, A317532, A326519, A382428.

Sequence in context: A368213 A249455 A379787 * A248967 A331094 A160169

Adjacent sequences: A382455 A382456 A382457 * A382459 A382460 A382461

Keyword

nonn,more

Author

Gus Wiseman, Mar 30 2025

Status

approved