A387329 Number of integer partitions of n such that it is not possible to choose a different constant integer partition of each part.

0, 0, 1, 1, 2, 3, 6, 8, 12, 18, 26, 35, 50, 67, 92, 122, 164, 214, 282, 364, 472, 604, 773, 978, 1240, 1555, 1953, 2432, 3027, 3744, 4629, 5687, 6982, 8533, 10414, 12655, 15364, 18580, 22442, 27018, 32485, 38943, 46624, 55671, 66389, 78980, 93836, 111244, 131717

Offset

0,5

Links

Max Alekseyev, Table of n, a(n) for n = 0..1000

Formula

G.f.: (1 - Product_{k>=1} (1 - x^(k*(A000005(k)+1)))) / Product_{k>=1} (1 - x^k). - Max Alekseyev, Nov 21 2025

Example

The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (211)   (311)    (222)     (511)      (611)
               (1111)  (2111)   (411)     (2221)     (2222)
                       (11111)  (2211)    (3211)     (3311)
                                (3111)    (4111)     (4211)
                                (21111)   (22111)    (5111)
                                (111111)  (31111)    (22211)
                                          (211111)   (32111)
                                          (1111111)  (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

Mathematica

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

Prog

(PARI) my(N=100, X=x+O(x^N)); Vec( (1-prod(i=1, N, 1-X^(i*(numdiv(i)+1)))) / prod(i=1, N, 1-X^i) ) \\ Max Alekseyev, Nov 21 2025

Crossrefs

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

For all (not just constant) partitions we have A387134, ranks A387577.

The complement all partitions is A387328, ranks A387576.

The complement strict partitions is A387178.

For strict (not just constant) partitions we have A387137.

These partitions are ranked by A387180.

The complement is A387330, ranked by A387181.

A000005 counts constant integer partitions.

A000009 counts strict integer partitions.

A000041 counts integer partitions.

Cf. A052335, A063834, A335448, A355739, A387110, A387115.

Sequence in context: A387578 A085642 A270738 * A049194 A058298 A299758

Adjacent sequences: A387326 A387327 A387328 * A387330 A387331 A387332

Keyword

nonn

Author

Gus Wiseman, Sep 05 2025

Extensions

Terms a(21) onward from Max Alekseyev, Nov 21 2025

Status

approved