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

Offset: 0

Author

Gus Wiseman, Mar 20 2025

Keywords

• nonn
• more

Examples

			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
		

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.

Programs

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}]

Extensions

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