A325329 Number of integer partitions of n whose multiplicities appear with distinct multiplicities.

1, 1, 2, 3, 4, 4, 8, 7, 13, 18, 25, 30, 52, 57, 81, 109, 140, 167, 230, 267, 354, 428, 532, 630, 815, 942, 1166, 1385, 1695, 1966, 2440, 2810, 3422, 4008, 4828, 5630, 6847, 7905, 9527, 11135, 13340, 15498, 18636, 21591, 25769, 30086, 35630, 41379, 49150, 56880

Offset: 0

Author

Gus Wiseman, May 01 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A325369.

Partitions whose parts appear with distinct multiplicities are counted by A098859, with Heinz numbers A130091.

Examples

			The a(0) = 1 through a(8) = 13 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (21)   (22)    (32)     (33)      (43)       (44)
                 (111)  (31)    (41)     (42)      (52)       (53)
                        (1111)  (11111)  (51)      (61)       (62)
                                         (222)     (421)      (71)
                                         (321)     (3211)     (431)
                                         (2211)    (1111111)  (521)
                                         (111111)             (2222)
                                                              (3221)
                                                              (3311)
                                                              (4211)
                                                              (32111)
                                                              (11111111)
For example, in (4,2,1,1), the multiplicities are 1 and 2, and 2 appears 1 time while 1 appears 2 times, so (4,2,1,1) is counted under a(8).
		

Crossrefs

Cf. A098859, A130091, A317081, A320348, A325326, A325330, A325331, A325333, A325337, A325369.

Programs

Mathematica

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