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A325279
Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.
1
0, 0, 0, 0, 1, 2, 1, 5, 6, 9, 10, 18, 18, 31, 34, 48, 57, 80, 86, 122, 138, 183, 211, 275, 311, 402, 461, 576, 663, 825, 942, 1163, 1334, 1621, 1865, 2248, 2566, 3084, 3532, 4193, 4794, 5674, 6472, 7617, 8685, 10153, 11576, 13483, 15320, 17790, 20200, 23342
OFFSET
0,6
COMMENTS
The Heinz numbers of these partitions are given by A325241.
For example, the partition (44111) has two multiplicities (2 and 3) which differ by 1, so is counted under a(11).
EXAMPLE
The a(4) = 1 through a(11) = 18 partitions:
(211) (221) (411) (322) (332) (441) (433) (443)
(311) (331) (422) (522) (442) (533)
(511) (611) (711) (622) (551)
(3211) (3221) (3321) (811) (722)
(22111) (4211) (4221) (5221) (911)
(22211) (4311) (5311) (4322)
(5211) (6211) (4331)
(32211) (33211) (4421)
(33111) (42211) (5411)
(2221111) (6221)
(6311)
(7211)
(33221)
(33311)
(43211)
(44111)
(52211)
(2222111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Max@@Length/@Split[#]-Min@@Length/@Split[#]==1&]], {n, 0, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 18 2019
STATUS
approved