A387330 Number of integer partitions of n such that it is possible to choose a different constant integer partition of each part.

1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 16, 21, 27, 34, 43, 54, 67, 83, 103, 126, 155, 188, 229, 277, 335, 403, 483, 578, 691, 821, 975, 1155, 1367, 1610, 1896, 2228, 2613, 3057, 3573, 4167, 4853, 5640, 6550, 7590, 8786, 10154, 11722, 13510, 15556, 17885, 20540

Offset

0,4

Comments

Also the number of integer partitions of n such that for each part k the multiplicity of k is at most A000005(k).

Links

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

Example

The partition (4,2,2,1) has choices such as ((2,2),(1,1),(2),(1)) so is counted under a(9).
The a(1) = 1 through a(9) = 12 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)   (432)
                                      (421)  (422)   (441)
                                             (431)   (522)
                                             (521)   (531)
                                             (3221)  (621)
                                                     (3321)
                                                     (4221)

Mathematica

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

Crossrefs

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

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

The complement for all partitions is A387134, ranks A387577.

The complement for strict partitions is A387137.

For strict instead of constant partitions we have A387178.

These partitions are ranked by A387181.

For all partitions (not just constant) we have A387328, ranks A387576.

The complement is counted by A387329, ranks A387180.

A000005 counts constant integer partitions.

A000009 counts strict integer partitions.

A000041 counts integer partitions.

A063834 counts twice-partitions.

Cf. A052335, A276079, A355731, A355739, A367771, A383706, A387110.

Sequence in context: A213267 A145977 A050729 * A117536 A104665 A094018

Adjacent sequences: A387327 A387328 A387329 * A387331 A387332 A387333

Keyword

nonn

Author

Gus Wiseman, Sep 07 2025

Status

approved