OFFSET
0,3
COMMENTS
Partitions with distinct multiplicities that cover an initial interval of positive integers are counted by A320348, with Heinz numbers A325337. Partitions whose multiplicities appear with distinct multiplicities are counted by A325329, with Heinz numbers A325369. Partitions whose multiplicities appear with multiplicities that cover an initial interval of positive integers of counted by A325330, with Heinz numbers A325370.
The Heinz numbers of these partitions are given by A325371.
EXAMPLE
The a(0) = 1 through a(8) = 7 partitions:
() (1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (11111) (33) (3211) (44)
(1111) (222) (1111111) (2222)
(111111) (3221)
(4211)
(32111)
(11111111)
For example, the partition p = (5,5,4,3,3,3,2,2) has multiplicities (2,3,1,2), which appear with multiplicities (1,2,1), which cover an initial interval but are not distinct, so p is not counted under a(27). The partition q = (5,5,5,4,4,4,3,3,2,2,1,1) has multiplicities (3,3,2,2,2), which appear with multiplicities (3,2), which are distinct but do not cover an initial interval, so q is not counted under a(39). The partition r = (3,3,2,1,1) has multiplicities (2,1,2), which appear with multiplicities (1,2), which are distinct and cover an initial interval, so r is counted under a(10).
MATHEMATICA
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n], normQ[Length/@Split[Sort[Length/@Split[#]]]]&&UnsameQ@@Length/@Split[Sort[Length/@Split[#]]]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 01 2019
STATUS
approved