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

0,3

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.

Links

Table of n, a(n) for n=0..19.

Example

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)

Mathematica

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

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.

Sequence in context: A033189 A008507 A318683 * A028364 A239482 A280470

Adjacent sequences: A383705 A383706 A383707 * A383709 A383710 A383711

Keyword

nonn,more

Author

Gus Wiseman, May 07 2025

Status

approved