A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 8, 8, 12, 17, 22, 29, 31, 40, 50, 65, 77, 101, 112, 135, 162, 201

Offset: 0

Author

Gus Wiseman, May 30 2025

Keywords

• nonn
• more

Comments

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

Examples

			For the partition (8,5,2) we have four choices:
  ((8),(4,1),(2))
  ((7,1),(5),(2))
  ((5,3),(4,1),(2))
  ((4,3,1),(5),(2))
Hence (8,5,2) is counted under a(15).
The a(5) = 1 through a(12) = 12 partitions:
  (5)  (6)    (7)  (8)    (9)    (10)     (11)     (12)
       (3,3)       (4,4)  (5,4)  (5,5)    (6,5)    (6,6)
                   (5,3)  (6,3)  (6,4)    (7,4)    (7,5)
                   (7,1)  (7,2)  (7,3)    (8,3)    (8,4)
                          (8,1)  (8,2)    (9,2)    (9,3)
                                 (9,1)    (10,1)   (10,2)
                                 (4,3,3)  (5,3,3)  (11,1)
                                 (4,4,2)  (5,5,1)  (5,5,2)
                                                   (6,3,3)
                                                   (6,4,2)
                                                   (6,5,1)
                                                   (9,2,1)
		

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 at least one proper choice we have A384317, ranked by A384321.

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

The strict version for just one proper choice is A384319, ranked by A384390.

For just one proper choice we have A384323, ranks A384347 = positions of 2 in A383706.

For no proper choices we have A384348, ranked by A384349.

These partitions are ranked by A384393.

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, non-strict A299200.

Cf. A098859, A279375, A317142, A381454, A382525, A382912, A382913, A384005.

Programs

Mathematica

pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pofprop[#]]>1&]],{n,0,15}]