cp's OEIS Frontend

A361907 Number of integer partitions of n such that (length) * (maximum) > 2*n.

%I A361907 #6 Apr 01 2023 22:03:47
%S A361907 0,0,0,0,0,0,3,4,7,11,19,26,43,60,80,115,171,201,297,374,485,656,853,
%T A361907 1064,1343,1758,2218,2673,3477,4218,5423,6523,7962,10017,12104,14409,
%U A361907 17978,22031,26318,31453,38176,45442,55137,65775,77451,92533,111485,131057
%N A361907 Number of integer partitions of n such that (length) * (maximum) > 2*n.
%C A361907 Also partitions such that (maximum) > 2*(mean).
%C A361907 These are partitions whose complement (see example) has size > n.
%e A361907 The a(7) = 3 through a(10) = 11 partitions:
%e A361907   (511)    (611)     (711)      (721)
%e A361907   (4111)   (5111)    (5211)     (811)
%e A361907   (31111)  (41111)   (6111)     (6211)
%e A361907            (311111)  (42111)    (7111)
%e A361907                      (51111)    (52111)
%e A361907                      (411111)   (61111)
%e A361907                      (3111111)  (421111)
%e A361907                                 (511111)
%e A361907                                 (3211111)
%e A361907                                 (4111111)
%e A361907                                 (31111111)
%e A361907 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 A361907 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 A361907 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 A361907 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 A361907 The partition y = (3,2,1,1) has diagram:
%e A361907   o o o
%e A361907   o o .
%e A361907   o . .
%e A361907   o . .
%e A361907 with complement (shown in dots) of size 5, and 5 is not > 7, so y is not counted under a(7).
%t A361907 Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#>2n&]],{n,30}]
%Y A361907 For length instead of mean we have A237751, reverse A237754.
%Y A361907 For minimum instead of mean we have A237820, reverse A053263.
%Y A361907 The complement is counted by A361851, median A361848.
%Y A361907 Reversing the inequality gives A361852.
%Y A361907 The equal version is A361853.
%Y A361907 For median instead of mean we have A361857, reverse A361858.
%Y A361907 Allowing equality gives A361906, median A361859.
%Y A361907 A000041 counts integer partitions, strict A000009.
%Y A361907 A008284 counts partitions by length, A058398 by mean.
%Y A361907 A051293 counts subsets with integer mean.
%Y A361907 A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A361907 A116608 counts partitions by number of distinct parts.
%Y A361907 A268192 counts partitions by complement size, ranks A326844.
%Y A361907 Cf. A027193, A111907, A237752, A237755, A237821, A237824, A237984, A324562, A326622, A327482, A349156, A360071, A361394.
%K A361907 nonn
%O A361907 1,7
%A A361907 _Gus Wiseman_, Mar 29 2023