A325333 -id:A325333 - cp's OEIS frontend

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

Gus Wiseman, May 01 2019

nonn

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.

			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).

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

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

A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 44, 55, 77, 96, 127, 158, 208, 251, 329, 400, 501, 610, 766, 915, 1141, 1368, 1677, 2005, 2454, 2913, 3553, 4219, 5110, 6053, 7300, 8644, 10376, 12238, 14645, 17216, 20504, 24047, 28501, 33336, 39373, 45871, 53926, 62745

Offset: 0

Gus Wiseman, May 01 2019

nonn

Partitions whose parts cover an initial interval of positive integers are counted by A000009, with Heinz numbers A055932. Partitions whose multiplicities cover an initial interval of positive integers are counted by A317081, with Heinz numbers A317090. Partitions whose parts and multiplicities both cover an initial interval of positive integers are counted by A317088, with Heinz numbers A317089. Partitions whose multiplicities at every depth cover an initial interval of positive integers are counted by A317245, with Heinz numbers A317246.

The Heinz numbers of these partitions are given by A325370.

			The a(0) = 1 through a(8) = 16 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (221)    (33)      (322)      (44)
                        (211)   (311)    (222)     (331)      (332)
                        (1111)  (2111)   (411)     (511)      (422)
                                (11111)  (3111)    (2221)     (611)
                                         (21111)   (3211)     (2222)
                                         (111111)  (4111)     (3221)
                                                   (22111)    (4211)
                                                   (31111)    (5111)
                                                   (211111)   (22211)
                                                   (1111111)  (32111)
                                                              (41111)
                                                              (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)
For example, the partition (5,5,4,3,3,3,2,2) has multiplicities (2,1,3,2) with multiplicities (1,2,1) which cover the initial interval {1,2}, so (5,5,4,3,3,3,2,2) is counted under a(27).

Cf. A000837, A055932, A317081, A317088, A317089, A317090, A317245, A320348, A325331, A325333, A325337, A325370.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&]],{n,0,30}]

cp's OEIS Frontend

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

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