A384323 Number of integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

0, 0, 0, 1, 1, 0, 3, 3, 2, 0, 6, 6, 6, 6, 4, 10, 10, 14, 16, 15, 16, 17, 20, 25, 27, 28, 37, 43, 31, 42, 44

Offset: 0

Author

Gus Wiseman, May 30 2025

Keywords

• nonn
• more

Examples

			For y = (4,3,3) we have two ways: ((4),(3),(2,1)) and ((4),(2,1),(3)), so y is counted under a(10).
The a(0) = 0 through a(15) = 10 partitions:
  .  .  .  3  4  .  33  43  44  .  433  533  543  544  554  5433
                    42  52  62     442  542  552  553  644  5442
                    51  61         532  551  633  652  662  5532
                                   541  632  732  661  833  5541
                                   631  731  741  733       6432
                                   721  821  831  832       6531
                                                            7431
                                                            7521
                                                            8421
                                                            9321
		

Crossrefs

For just one choice we have A179009, ranked by A383707.

Twice-partitions of this type are counted by A279790.

For at least one choice we have A383708, odd case A383533.

For no choices we have A383710, odd case A383711.

For more than one choice we have A384317, ranked by A384321.

The strict version for at least one choice is A384318, ranked by A384322.

The strict version is A384319, ranked by A384390.

These partitions are ranked by A384347 = positions of 2 in A383706.

A000041 counts integer partitions, strict A000009.

A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.

A239455 counts Look-and-Say partitions, ranks A351294 or A381432.

A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.

A357982 counts choices of strict partitions of each prime index.

Cf. A098859, A299200, A317142, A381454, A382525, A382912, A382913, A384005.

Programs

Mathematica

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pof[#]]==2&]],{n,0,15}]