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A325244
Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.
11
0, 0, 0, 1, 1, 2, 3, 4, 7, 12, 16, 21, 33, 38, 50, 75, 87, 111, 150, 185, 244, 307, 373, 461, 585, 702, 856, 1043, 1255, 1498, 1822, 2143, 2565, 3064, 3607, 4251, 5064, 5920, 6953, 8174, 9503, 11064, 12927, 14921, 17320, 19986, 23067, 26485, 30499, 34894
OFFSET
0,6
COMMENTS
For example, (32211) has two distinct multiplicities (1, 2) and three distinct parts (1, 2, 3) so is counted under a(9).
The Heinz numbers of these partitions are given by A325259.
EXAMPLE
The a(3) = 1 through a(10) = 16 partitions:
(21) (31) (32) (42) (43) (53) (54) (64)
(41) (51) (52) (62) (63) (73)
(2211) (61) (71) (72) (82)
(3211) (3221) (81) (91)
(3311) (3321) (3322)
(4211) (4221) (4411)
(32111) (4311) (5221)
(5211) (5311)
(32211) (6211)
(42111) (32221)
(222111) (33211)
(321111) (42211)
(43111)
(52111)
(421111)
(3211111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Union[#]]==Length[Union[Length/@Split[#]]]+1&]], {n, 0, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2019
STATUS
approved