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Number of integer partitions of n such that (length) * (maximum) < 2n.
8

%I #7 Mar 31 2023 05:01:30

%S 1,2,3,5,7,9,12,17,21,27,37,41,58,67,80,106,126,153,193,209,263,326,

%T 402,419,565,650,694,891,1088,1120,1419,1672,1987,2245,2345,2856,3659,

%U 3924,4519,4975,6407,6534,8124,8280,9545,12937,13269,13788,16474,20336

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

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

%e The a(1) = 1 through a(7) = 12 partitions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (11) (21) (22) (32) (33) (43)

%e (111) (31) (41) (42) (52)

%e (211) (221) (51) (61)

%e (1111) (311) (222) (322)

%e (2111) (321) (331)

%e (11111) (2211) (421)

%e (21111) (2221)

%e (111111) (3211)

%e (22111)

%e (211111)

%e (1111111)

%e 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).

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

%Y For length instead of mean we have A237754.

%Y Allowing equality gives A237755, for median A361848.

%Y For equal median we have A361849, ranks A361856.

%Y The equal version is A361853, ranks A361855.

%Y For median instead of mean we have A361858.

%Y The complement is counted by A361906.

%Y Reversing the inequality gives A361907.

%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.

%Y Cf. A027193, A111907, A116608, A237824, A237984, A324517, A327482, A349156, A360068, A360071, A361394.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 29 2023