A382079 - OEIS

A382079

Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 9, 13, 14, 21, 20, 32, 31, 42, 47, 63, 62, 90, 94, 117, 138, 170, 186, 235, 260, 315, 363, 429, 493, 588, 674, 795, 901, 1060, 1209, 1431, 1608, 1896, 2152, 2515, 2854, 3310, 3734, 4368, 4905, 5686

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OFFSET

0,5

LINKS

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

EXAMPLE

The unique multiset partition for (3222111) is {{1},{2},{1,2},{1,2,3}}.

The a(1) = 1 through a(12) = 13 partitions:

1 2 3 4 5 6 7 8 9 A B C

211 221 411 322 332 441 433 443 552

311 2211 331 422 522 442 533 633

511 611 711 622 551 822

3311 42111 811 722 A11

32111 3322 911 4422

4411 42221 5511

32221 53111 33321

43111 62111 52221

52111 54111

63111

72111

3222111

MATHEMATICA

ssfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#, d]&)/@Select[ssfacs[n/d], Min@@#>d&], {d, Select[Rest[Divisors[n]], SquareFreeQ]}]];

Table[Length[Select[IntegerPartitions[n], Length[ssfacs[Times@@Prime/@#]]==1&]], {n, 0, 15}]

CROSSREFS

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

These partitions are ranked by A293511.

MM-numbers of these multiset partitions (sets of sets) are A302494, see A302478, A382201.

Twice-partitions of this type (sets of sets) are counted by A358914, see A279785.

For at least one choice we have A382077 (ranks A382200), see A381992 (ranks A382075).

For no choices we have A382078 (ranks A293243), see A381990 (ranks A381806).

For distinct block-sums instead of blocks we have A382460, ranked by A381870.

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

A000041 counts integer partitions, strict A000009.

A050320 counts multiset partitions of prime indices into sets.

A050326 counts multiset partitions of prime indices into distinct sets, see A381633.

A265947 counts refinement-ordered pairs of integer partitions.

Cf. A002846, A213427, A299202, A317142, A381078, A381441, A381454, A381636.

Sequence in context: A381048 A131187 A099072 * A257241 A382460 A239964

Adjacent sequences: A382076 A382077 A382078 * A382080 A382081 A382082

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Mar 20 2025

EXTENSIONS

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

STATUS

approved