A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

0, 0, 1, 1, 2, 3, 6, 8, 12, 17, 25, 34, 49, 65, 89, 118, 158, 206, 271, 349, 453, 578, 740, 935, 1186, 1486, 1865, 2322, 2890, 3572, 4415, 5423, 6659, 8134, 9927, 12062, 14643, 17706, 21387, 25746, 30957, 37109, 44433, 53054, 63273, 75276, 89444, 106044

Offset

0,5

Comments

Number of integer partitions of n such that it is not possible to choose a sequence of distinct integer partitions, one of each part.

Also the number of integer partitions of n with at least one part k satisfying that the multiplicity of k exceeds the number of integer partitions of k.

Links

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

Formula

G.f.: (1 - Product_{k>=1} (1 - x^(k*(A000041(k)+1)))) / Product_{k>=1} (1 - x^k). - Max Alekseyev, Nov 20 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

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

Crossrefs

These partitions are ranked by A387577 (not A276079).

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

The complement for prime factors is A370592, ranks A368100.

For prime factors instead of partitions we have A370593, ranks A355529.

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

For just choices of strict partitions we have A387137.

The complement is counted by A387328, ranks A387576 (not A276078).

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

Cf. A335433, A355535, A367867, A367901, A367903, A367905, A367907, A370583, A370594.

Sequence in context: A343820 A133582 A325342 * A387578 A085642 A270738

Adjacent sequences: A387131 A387132 A387133 * A387135 A387136 A387137

Keyword

nonn

Author

Gus Wiseman, Aug 29 2025

Status

approved