A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 34, 47

Offset

0,4

Comments

First differs from A382523 at a(12) = 47, A382523(12) = 45.

We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.

Links

Table of n, a(n) for n=0..12.

Example

Differs from A382523 in counting the following under a(12):
  {1,1,1,1,1,1,2,2,3,3,4,5} with partition {{1},{1,2},{1,3},{1,4},{1,5},{1,2,3}}
  {1,1,1,1,2,2,2,2,3,3,3,3} with partition {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Mathematica

strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]& /@ IntegerPartitions[n];
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]]]];
Table[Length[Select[strnorm[n], Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]], {n, 0, 5}]

Crossrefs

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

Normal multiset partitions of this type are counted by A116539, distinct sums A381718.

The complement is counted by A292444.

Twice-partitions of this type are counted by A358914, distinct sums A279785.

For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.

Weak version is A382214, complement A292432, distinct sums A382216, complement A382202.

For distinct sums we have A382523, complement A382430.

Normal multiset partitions: A034691, A035310, A116540, A255906.

Set systems: A050342, A296120, A318361.

Set multipartitions: A089259, A270995, A296119, A318360.

Cf. A000110, A000670, A007716, A050320, A255903, A317532, A326519, A381992, A382428, A382458, A382459.

Sequence in context: A016028 A239551 A382523 * A384884 A219282 A098578

Adjacent sequences: A381993 A381994 A381995 * A381997 A381998 A381999

Keyword

nonn,more

Author

Gus Wiseman, Mar 31 2025

Status

approved