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

Author

Gus Wiseman, Aug 29 2025

Keywords

• nonn
• new

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.

Examples

			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)
		

Crossrefs

These partitions are ranked by 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 A276078.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

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

Programs

Mathematica

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