A336866 Number of integer partitions of n without all distinct multiplicities.

0, 0, 0, 1, 1, 2, 4, 5, 9, 15, 21, 28, 46, 56, 80, 114, 149, 192, 269, 337, 455, 584, 751, 943, 1234, 1527, 1944, 2422, 3042, 3739, 4699, 5722, 7100, 8668, 10634, 12880, 15790, 19012, 23093, 27776, 33528, 40102, 48264, 57469, 68793, 81727, 97372, 115227

Offset

0,6

Links

Table of n, a(n) for n=0..47.

Formula

a(n) = A000041(n) - A098859(n).

Example

The a(0) = 0 through a(9) = 15 partitions (empty columns shown as dots):
  .  .  .  (21)  (31)  (32)  (42)    (43)    (53)     (54)
                       (41)  (51)    (52)    (62)     (63)
                             (321)   (61)    (71)     (72)
                             (2211)  (421)   (431)    (81)
                                     (3211)  (521)    (432)
                                             (3221)   (531)
                                             (3311)   (621)
                                             (4211)   (3321)
                                             (32111)  (4221)
                                                      (4311)
                                                      (5211)
                                                      (32211)
                                                      (42111)
                                                      (222111)
                                                      (321111)

Mathematica

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&]], {n, 0, 30}]

Crossrefs

A098859 counts the complement.

A130092 gives the Heinz numbers of these partitions.

A001222 counts prime factors with multiplicity.

A013929 lists nonsquarefree numbers.

A047966 counts uniform partitions.

A047967 counts non-strict partitions.

A071625 counts distinct prime multiplicities.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A327498 gives the maximum divisor with distinct prime multiplicities.

Cf. A000009, A000041, A008284, A116608, A118914, A124010, A242882, A255231, A325280, A325242, A329739, A336422, A336571.

Sequence in context: A120770 A266990 A349738 * A255213 A226447 A073151

Adjacent sequences: A336863 A336864 A336865 * A336867 A336868 A336869

Keyword

nonn

Author

Gus Wiseman, Aug 09 2020

Status

approved