cp's OEIS Frontend

A361852 Number of integer partitions of n such that (length) * (maximum) < 2n.

%I A361852 #7 Mar 31 2023 05:01:30
%S A361852 1,2,3,5,7,9,12,17,21,27,37,41,58,67,80,106,126,153,193,209,263,326,
%T A361852 402,419,565,650,694,891,1088,1120,1419,1672,1987,2245,2345,2856,3659,
%U A361852 3924,4519,4975,6407,6534,8124,8280,9545,12937,13269,13788,16474,20336
%N A361852 Number of integer partitions of n such that (length) * (maximum) < 2n.
%C A361852 Also partitions such that (maximum) < 2*(mean).
%e A361852 The a(1) = 1 through a(7) = 12 partitions:
%e A361852   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A361852        (11)  (21)   (22)    (32)     (33)      (43)
%e A361852              (111)  (31)    (41)     (42)      (52)
%e A361852                     (211)   (221)    (51)      (61)
%e A361852                     (1111)  (311)    (222)     (322)
%e A361852                             (2111)   (321)     (331)
%e A361852                             (11111)  (2211)    (421)
%e A361852                                      (21111)   (2221)
%e A361852                                      (111111)  (3211)
%e A361852                                                (22111)
%e A361852                                                (211111)
%e A361852                                                (1111111)
%e A361852 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 A361852 Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#<2n&]],{n,30}]
%Y A361852 For length instead of mean we have A237754.
%Y A361852 Allowing equality gives A237755, for median A361848.
%Y A361852 For equal median we have A361849, ranks A361856.
%Y A361852 The equal version is A361853, ranks A361855.
%Y A361852 For median instead of mean we have A361858.
%Y A361852 The complement is counted by A361906.
%Y A361852 Reversing the inequality gives A361907.
%Y A361852 A000041 counts integer partitions, strict A000009.
%Y A361852 A008284 counts partitions by length, A058398 by mean.
%Y A361852 A051293 counts subsets with integer mean.
%Y A361852 A067538 counts partitions with integer mean.
%Y A361852 Cf. A027193, A111907, A116608, A237824, A237984, A324517, A327482, A349156, A360068, A360071, A361394.
%K A361852 nonn
%O A361852 1,2
%A A361852 _Gus Wiseman_, Mar 29 2023