cp's OEIS Frontend

A361851 Number of integer partitions of n such that (length) * (maximum) <= 2*n.

%I A361851 #9 Mar 31 2023 05:01:37
%S A361851 1,2,3,5,7,11,12,18,23,31,37,51,58,75,96,116,126,184,193,253,307,346,
%T A361851 402,511,615,678,792,1045,1088,1386,1419,1826,2181,2293,2779,3568,
%U A361851 3659,3984,4867,5885,6407,7732,8124,9400,11683,13025,13269,16216,17774,22016
%N A361851 Number of integer partitions of n such that (length) * (maximum) <= 2*n.
%C A361851 Also partitions such that (maximum) <= 2*(mean).
%C A361851 These are partitions whose complement (see example) has size <= n.
%e A361851 The a(1) = 1 through a(7) = 12 partitions:
%e A361851   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A361851        (11)  (21)   (22)    (32)     (33)      (43)
%e A361851              (111)  (31)    (41)     (42)      (52)
%e A361851                     (211)   (221)    (51)      (61)
%e A361851                     (1111)  (311)    (222)     (322)
%e A361851                             (2111)   (321)     (331)
%e A361851                             (11111)  (411)     (421)
%e A361851                                      (2211)    (2221)
%e A361851                                      (3111)    (3211)
%e A361851                                      (21111)   (22111)
%e A361851                                      (111111)  (211111)
%e A361851                                                (1111111)
%e A361851 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).
%e A361851 The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 is not <= 2*9, so y is not counted under a(9).
%e A361851 The partition y = (3,2,1,1) has diagram:
%e A361851   o o o
%e A361851   o o .
%e A361851   o . .
%e A361851   o . .
%e A361851 with complement of size 5, and 5 <= 7, so y is counted under a(7).
%t A361851 Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#<=2n&]],{n,30}]
%Y A361851 For length instead of mean we have A237755.
%Y A361851 For minimum instead of mean we have A237824.
%Y A361851 For median instead of mean we have A361848.
%Y A361851 The equal case for median is A361849, ranks A361856.
%Y A361851 The unequal case is A361852, median A361858.
%Y A361851 The equal case is A361853, ranks A361855.
%Y A361851 Reversing the inequality gives A361906, unequal case A361907.
%Y A361851 A000041 counts integer partitions, strict A000009.
%Y A361851 A008284 counts partitions by length, A058398 by mean.
%Y A361851 A051293 counts subsets with integer mean.
%Y A361851 A067538 counts partitions with integer mean.
%Y A361851 Cf. A111907, A237984, A240219, A324521, A324562, A327482, A349156, A360068, A360071, A360241, A361394, A361859.
%K A361851 nonn
%O A361851 1,2
%A A361851 _Gus Wiseman_, Mar 28 2023