A360242 - OEIS

%I #9 Feb 06 2023 10:06:20

%S 0,0,0,0,1,3,3,9,11,19,25,43,49,82,103,136,183,258,314,435,524,687,

%T 892,1150,1378,1788,2241,2773,3399,4308,5142,6501,7834,9600,11726,

%U 14099,16949,20876,25042,30032,35732,43322,51037,61650,72807,86319,102983,122163

%N Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

%e The a(1) = 0 through a(9) = 19 partitions:

%e   .  .  .  (211)  (221)   (411)    (322)     (332)      (441)

%e                   (311)   (3111)   (331)     (422)      (522)

%e                   (2111)  (21111)  (511)     (611)      (711)

%e                                    (2221)    (4211)     (3222)

%e                                    (3211)    (5111)     (3321)

%e                                    (4111)    (22211)    (4221)

%e                                    (22111)   (32111)    (4311)

%e                                    (31111)   (41111)    (5211)

%e                                    (211111)  (221111)   (6111)

%e                                              (311111)   (22221)

%e                                              (2111111)  (32211)

%e                                                         (33111)

%e                                                         (42111)

%e                                                         (51111)

%e                                                         (321111)

%e                                                         (411111)

%e                                                         (2211111)

%e                                                         (3111111)

%e                                                         (21111111)

%e For example, the partition y = (32211) has mean 9/5 and distinct parts {1,2,3} with mean 2, so y is counted under a(9).

%t Table[Length[Select[IntegerPartitions[n],Mean[#]!=Mean[Union[#]]&]],{n,0,30}]

%Y The complement for multiplicities instead of distinct parts is A360068.

%Y The complement is counted by A360243, ranks A360247.

%Y For median instead of mean we have A360244, complement A360245.

%Y These partitions have ranks A360246.

%Y Sum of A360250 and A360251, ranks A360252 and A360253.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by number of parts.

%Y A058398 counts partitions by mean, also A327482.

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

%Y A116608 counts partitions by number of distinct parts.

%Y A360071 counts partitions by number of parts and number of distinct parts.

%Y A360241 counts partitions whose distinct parts have integer mean.

%Y Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156.

%K nonn

%O 0,6

%A _Gus Wiseman_, Feb 04 2023