A387578 Number of integer partitions of n such that for some part k the multiplicity of k exceeds k.

0, 0, 1, 1, 2, 3, 6, 8, 12, 17, 25, 34, 49, 65, 89, 118, 158, 206, 271, 349, 454, 579, 741, 937, 1188, 1489, 1869, 2327, 2897, 3581, 4428, 5439, 6678, 8159, 9959, 12102, 14693, 17768, 21465, 25842, 31074, 37253, 44609, 53268, 63533, 75591, 89825, 106502, 126125

Offset

0,5

Comments

The Heinz numbers of these partitions are given by A276079.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Example

The partition (2,2,2,1) has more than two 2's so is counted under a(7).
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)

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t, add(
      b(n-i*j, i-1, `if`(j>i, 1, t)), j=`if`(i=1, n, 0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..48);  # Alois P. Heinz, Oct 07 2025

Mathematica

Table[Length[Select[IntegerPartitions[n], Or@@Table[Count[#, k]>k, {k, Union[#]}]&]], {n, 0, 30}]

Crossrefs

The complement is counted by A052335, ranks A276078.

For initial intervals we have A238873 (ranks A387112), A387118 (ranks A387113).

These partitions have ranks A276079.

For divisors we have A387329, ranks A387180.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A276429 counts partitions with no part k of multiplicity k, ranks A325130.

A367902 counts choosable set-systems, complement A367903.

A370582 counts sets with choosable prime factors, complement A370583.

Cf. A087153, A117144, A239312, A367867, A370592, A370593, A370594, A387111, A387134, A387328, A387178.

Sequence in context: A133582 A325342 A387134 * A085642 A270738 A387329

Adjacent sequences: A387575 A387576 A387577 * A387579 A387580 A387581

Keyword

nonn

Author

Gus Wiseman, Sep 10 2025

Status

approved