%I #6 Apr 01 2023 22:03:47

%S 0,0,0,0,0,0,3,4,7,11,19,26,43,60,80,115,171,201,297,374,485,656,853,

%T 1064,1343,1758,2218,2673,3477,4218,5423,6523,7962,10017,12104,14409,

%U 17978,22031,26318,31453,38176,45442,55137,65775,77451,92533,111485,131057

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

%C Also partitions such that (maximum) > 2*(mean).

%C These are partitions whose complement (see example) has size > n.

%e The a(7) = 3 through a(10) = 11 partitions:

%e (511) (611) (711) (721)

%e (4111) (5111) (5211) (811)

%e (31111) (41111) (6111) (6211)

%e (311111) (42111) (7111)

%e (51111) (52111)

%e (411111) (61111)

%e (3111111) (421111)

%e (511111)

%e (3211111)

%e (4111111)

%e (31111111)

%e 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).

%e The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 is not > 2*8, so y is not counted under a(8).

%e The partition y = (5,1,1,1) has length 4 and maximum 5, and 4*5 > 2*8, so y is counted under a(8).

%e The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 > 2*9, so y is counted under a(9).

%e The partition y = (3,2,1,1) has diagram:

%e o o o

%e o o .

%e o . .

%e o . .

%e with complement (shown in dots) of size 5, and 5 is not > 7, so y is not counted under a(7).

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

%Y For length instead of mean we have A237751, reverse A237754.

%Y For minimum instead of mean we have A237820, reverse A053263.

%Y The complement is counted by A361851, median A361848.

%Y Reversing the inequality gives A361852.

%Y The equal version is A361853.

%Y For median instead of mean we have A361857, reverse A361858.

%Y Allowing equality gives A361906, median A361859.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, A058398 by mean.

%Y A051293 counts subsets with integer mean.

%Y A067538 counts partitions with integer mean, strict A102627, ranks A316413.

%Y A116608 counts partitions by number of distinct parts.

%Y A268192 counts partitions by complement size, ranks A326844.

%Y Cf. A027193, A111907, A237752, A237755, A237821, A237824, A237984, A324562, A326622, A327482, A349156, A360071, A361394.

%K nonn

%O 1,7

%A _Gus Wiseman_, Mar 29 2023