

A361906


Number of integer partitions of n such that (length) * (maximum) >= 2*n.


10



0, 0, 0, 0, 0, 2, 3, 5, 9, 15, 19, 36, 43, 68, 96, 125, 171, 232, 297, 418, 529, 676, 853, 1156, 1393, 1786, 2316, 2827, 3477, 4484, 5423, 6677, 8156, 10065, 12538, 15121, 17978, 22091, 26666, 32363, 38176, 46640, 55137, 66895, 79589, 92621, 111485, 133485
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OFFSET

1,6


COMMENTS

Also partitions such that (maximum) >= 2*(mean).
These are partitions whose complement (see example) has size >= n.


LINKS



EXAMPLE

The a(6) = 2 through a(10) = 15 partitions:
(411) (511) (611) (621) (721)
(3111) (4111) (4211) (711) (811)
(31111) (5111) (5211) (5221)
(41111) (6111) (5311)
(311111) (42111) (6211)
(51111) (7111)
(321111) (42211)
(411111) (43111)
(3111111) (52111)
(61111)
(421111)
(511111)
(3211111)
(4111111)
(31111111)
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 >= 2*8, so y is counted under a(8).
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 is not >= 2*7, so y is not counted under a(7).
The partition y = (3,2,1,1) has diagram:
o o o
o o .
o . .
o . .
with complement (shown in dots) of size 5, and 5 is not >= 7, so y is not counted under a(7).


MATHEMATICA

Table[Length[Select[IntegerPartitions[n], Length[#]*Max@@#>=2n&]], {n, 30}]


CROSSREFS

The complement is counted by A361852.
Reversing the inequality gives A361851.
A051293 counts subsets with integer mean.


KEYWORD

nonn


AUTHOR



STATUS

approved



