A384348 Number of integer partitions of n with no proper way to choose disjoint strict partitions of each part.

1, 1, 2, 2, 4, 6, 7, 11, 17, 25, 30, 44, 61, 82, 113, 141, 193, 249, 327, 422, 548, 682, 881, 1106, 1400, 1751

Offset: 0

Author

Gus Wiseman, May 30 2025

Keywords

• nonn

Comments

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

Examples

			For the partition y = (5,4,2,1) we have the following proper ways to choose strict partitions of each part:
  ((5),(3,1),(2),(1))
  ((4,1),(4,2),(1))
  ((4,1),(3,1),(2),(1))
  ((3,2),(4),(2),(1))
  ((3,2),(3,1),(2),(1))
But none of this is disjoint, so y is counted under a(12).
The a(1) = 1 through a(8) = 17 partitions:
  (1)  (2)   (21)   (22)    (32)     (222)     (322)      (332)
       (11)  (111)  (31)    (41)     (321)     (331)      (422)
                    (211)   (221)    (411)     (421)      (431)
                    (1111)  (311)    (2211)    (511)      (521)
                            (2111)   (3111)    (2221)     (611)
                            (11111)  (21111)   (3211)     (2222)
                                     (111111)  (4111)     (3221)
                                               (22111)    (3311)
                                               (31111)    (4211)
                                               (211111)   (5111)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

The strict case is A179009, ranked by A383707.

This is the proper version of A383710, odd case A383711.

This is the proper complement of A383708, odd case A383533.

The complement is counted by A384317, ranks A384321.

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

For just one proper choice we have A384319, ranked by A384390.

For two choices we have A384323, ranks A384347 = positions of 2 in A383706.

These partitions are ranked by A384349.

For more than one proper choice we have A384395, 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.

Cf. A098859, A279790, A299200, A317142, A357982, A381454, A382525, A382912, A382913, A384005.

Programs

Mathematica

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