A361857 Number of integer partitions of n such that the maximum is greater than twice the median.

0, 0, 0, 0, 1, 2, 3, 7, 11, 16, 25, 37, 52, 74, 101, 138, 185, 248, 325, 428, 554, 713, 914, 1167, 1476, 1865, 2336, 2922, 3633, 4508, 5562, 6854, 8405, 10284, 12536, 15253, 18489, 22376, 26994, 32507, 39038, 46802, 55963, 66817, 79582, 94643, 112315

Offset

1,6

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..47.

Example

The a(5) = 1 through a(10) = 16 partitions:
  (311)  (411)   (511)    (521)     (522)      (622)
         (3111)  (4111)   (611)     (621)      (721)
                 (31111)  (4211)    (711)      (811)
                          (5111)    (5211)     (5221)
                          (32111)   (6111)     (5311)
                          (41111)   (33111)    (6211)
                          (311111)  (42111)    (7111)
                                    (51111)    (43111)
                                    (321111)   (52111)
                                    (411111)   (61111)
                                    (3111111)  (331111)
                                               (421111)
                                               (511111)
                                               (3211111)
                                               (4111111)
                                               (31111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 > 2*2, so y is counted under a(10).

Mathematica

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

Crossrefs

For length instead of median we have A237751.

For minimum instead of median we have A237820.

The complement is counted by A361848.

The equal version is A361849, ranks A361856.

Reversing the inequality gives A361858.

Allowing equality gives A361859, ranks A361868.

These partitions have ranks A361867.

For mean instead of median we have A361907.

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, A013580, A027193, A061395, A237755, A237824, A240219, A361394, A361851, A361860.

Sequence in context: A100963 A342029 A368403 * A197636 A347756 A040103

Adjacent sequences: A361854 A361855 A361856 * A361858 A361859 A361860

Keyword

nonn

Author

Gus Wiseman, Apr 02 2023

Status

approved