A387328 Number of integer partitions of n whose parts have choosable sets of integer partitions.

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 17, 22, 28, 36, 46, 58, 73, 91, 114, 141, 174, 214, 262, 320, 389, 472, 571, 688, 828, 993, 1189, 1419, 1690, 2009, 2383, 2821, 3334, 3931, 4628, 5439, 6381, 7474, 8741, 10207, 11902, 13858, 16114, 18710, 21698, 25130, 29070

Offset

0,4

Comments

First differs from A052335 at A052335(20) = 173, a(20) = 174, corresponding to the partition (4,4,4,4,4).

a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct integer partitions, one of each part.

Also the number of integer partitions y of n with no part k whose multiplicity in y exceeds A000041(k).

Links

Table of n, a(n) for n=0..50.

Example

The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
            (21)  (22)  (32)   (33)   (43)   (44)    (54)
                  (31)  (41)   (42)   (52)   (53)    (63)
                        (221)  (51)   (61)   (62)    (72)
                               (321)  (322)  (71)    (81)
                                      (331)  (332)   (333)
                                      (421)  (422)   (432)
                                             (431)   (441)
                                             (521)   (522)
                                             (3221)  (531)
                                                     (621)
                                                     (3321)
                                                     (4221)

Mathematica

Table[Length[Select[IntegerPartitions[n], Select[Tuples[IntegerPartitions/@#], UnsameQ@@#&]!={}&]], {n, 0, 15}]

Crossrefs

The strict case is A000009.

For initial intervals instead of partitions we have A238873, complement A387118.

For divisors instead of partitions we have A239312, complement A370320.

For prime factors instead of partitions we have A370592, ranks A368100.

The complement for prime factors is A370593, ranks A355529.

The complement is counted by A387134, ranks A387577.

For sets of strict partitions we have A387178, complement A387137.

These partitions are ranked by A387576.

A000005 counts divisors.

A000041 counts integer partitions.

Cf. A052335, A052337, A335433, A367771, A370585, A370804.

Sequence in context: A106507 A006950 A052335 * A389642 A193771 A160333

Adjacent sequences: A387325 A387326 A387327 * A387329 A387330 A387331

Keyword

nonn

Author

Gus Wiseman, Sep 01 2025

Status

approved