A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

1, 1, 2, 2, 3, 5, 5, 7, 8, 13, 14, 18, 22, 27, 36, 41, 50, 61, 73, 86

Offset: 0

Author

Gus Wiseman, May 07 2025

Keywords

• nonn
• more

Comments

Also the number of integer partitions y of n whose normal multiset (in which i appears y_i times) is a Look-and-Say partition.

Examples

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is counted under a(6).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (4,2,1)  (7,1)    (8,1)
                                                   (4,3,1)  (4,3,2)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
		

Crossrefs

These partitions have Heinz numbers A382913.

Without ones we have A383533, complement A383711.

The number of such families for each Heinz number is A383706.

The complement is counted by A383710, ranks A382912.

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

A098859 counts partitions with distinct multiplicities, compositions A242882.

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

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

Cf. A044813, A047966, A089259, A116540, A091602, A130091, A317141, A351013, A381441, A382771, A383013.

Programs

Mathematica

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