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A361852
Number of integer partitions of n such that (length) * (maximum) < 2n.
8
1, 2, 3, 5, 7, 9, 12, 17, 21, 27, 37, 41, 58, 67, 80, 106, 126, 153, 193, 209, 263, 326, 402, 419, 565, 650, 694, 891, 1088, 1120, 1419, 1672, 1987, 2245, 2345, 2856, 3659, 3924, 4519, 4975, 6407, 6534, 8124, 8280, 9545, 12937, 13269, 13788, 16474, 20336
OFFSET
1,2
COMMENTS
Also partitions such that (maximum) < 2*(mean).
EXAMPLE
The a(1) = 1 through a(7) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (32) (33) (43)
(111) (31) (41) (42) (52)
(211) (221) (51) (61)
(1111) (311) (222) (322)
(2111) (321) (331)
(11111) (2211) (421)
(21111) (2221)
(111111) (3211)
(22111)
(211111)
(1111111)
For example, the partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 < 2*7, so y is counted under a(7).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#]*Max@@#<2n&]], {n, 30}]
CROSSREFS
For length instead of mean we have A237754.
Allowing equality gives A237755, for median A361848.
For equal median we have A361849, ranks A361856.
The equal version is A361853, ranks A361855.
For median instead of mean we have A361858.
The complement is counted by A361906.
Reversing the inequality gives A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
Sequence in context: A351874 A080000 A333573 * A032459 A263647 A028870
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 29 2023
STATUS
approved