A325705 Number of integer partitions of n containing all of their distinct multiplicities.

1, 1, 0, 1, 3, 2, 4, 3, 7, 8, 16, 15, 24, 28, 39, 44, 68, 80, 98, 130, 167, 200, 259, 320, 396, 497, 601, 737, 910, 1107, 1335, 1631, 1983, 2372, 2887, 3439, 4166, 4949, 5940, 7043, 8450, 9980, 11884, 13984, 16679, 19493, 23162, 27050, 31937, 37334, 43926

Offset

0,5

Comments

The Heinz numbers of these partitions are given by A325706.

Links

Table of n, a(n) for n=0..50.

Example

The partition (4,2,1,1,1,1) has distinct multiplicities {1,4}, both of which belong to the partition, so it is counted under a(10).
The a(0) = 1 through a(10) = 16 partitions:
  ()  (1)  (21)  (22)   (41)   (51)    (61)    (71)     (81)     (91)
                 (31)   (221)  (321)   (421)   (431)    (333)    (541)
                 (211)         (2211)  (3211)  (521)    (531)    (631)
                               (3111)          (3221)   (621)    (721)
                                               (4211)   (3321)   (3322)
                                               (32111)  (4221)   (3331)
                                               (41111)  (5211)   (4321)
                                                        (32211)  (5221)
                                                                 (6211)
                                                                 (32221)
                                                                 (33211)
                                                                 (42211)
                                                                 (43111)
                                                                 (322111)
                                                                 (421111)
                                                                 (511111)

Mathematica

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

Crossrefs

Cf. A109297, A114639, A114640, A181819, A225486, A290689, A324753, A324843, A325702, A325706, A325707, A325755.

Sequence in context: A070961 A243291 A245962 * A243407 A061721 A294209

Adjacent sequences: A325702 A325703 A325704 * A325706 A325707 A325708

Keyword

nonn

Author

Gus Wiseman, May 18 2019

Status

approved