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

0,3

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.

Links

Table of n, a(n) for n=0..49.

Example

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).

Mathematica

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

Crossrefs

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

Sequence in context: A360245 A364674 A319079 * A224038 A344680 A330147

Adjacent sequences: A325326 A325327 A325328 * A325330 A325331 A325332

Keyword

nonn

Author

Gus Wiseman, May 01 2019

Status

approved