A361858 Number of integer partitions of n such that the maximum is less than twice the median.

1, 2, 3, 4, 5, 8, 8, 12, 15, 19, 22, 31, 34, 45, 55, 67, 78, 100, 115, 144, 170, 203, 238, 291, 337, 403, 473, 560, 650, 772, 889, 1046, 1213, 1414, 1635, 1906, 2186, 2533, 2913, 3361, 3847, 4433, 5060, 5808, 6628, 7572, 8615, 9835, 11158, 12698, 14394

Offset

1,2

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Links

Table of n, a(n) for n=1..51.

Example

The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (322)      (71)
                                     (321)     (331)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
The partition y = (3,2,2,1) has maximum 3 and median 2, and 3 < 2*2, so y is counted under a(8).

Mathematica

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

Crossrefs

For minimum instead of median we have A053263.

For length instead of median we have A237754.

Allowing equality gives A361848, strict A361850.

The equal version is A361849, ranks A361856.

For mean instead of median we have A361852.

Reversing the inequality gives A361857, ranks A361867.

The complement is counted by A361859, ranks A361868.

A000041 counts integer partitions, strict A000009.

A000975 counts subsets with integer median.

A325347 counts partitions with integer median, complement A307683.

A359893 and A359901 count partitions by median.

A360005 gives twice median of prime indices, distinct A360457.

Cf. A008284, A027193, A237751, A237755, A237820, A237824, A240219, A361394, A361851, A361860, A361907.

Sequence in context: A332339 A100882 A171979 * A181694 A364919 A297166

Adjacent sequences: A361855 A361856 A361857 * A361859 A361860 A361861

Keyword

nonn

Author

Gus Wiseman, Apr 02 2023

Status

approved