A361859 Number of integer partitions of n such that the maximum is greater than or equal to twice the median.

0, 0, 0, 1, 2, 3, 7, 10, 15, 23, 34, 46, 67, 90, 121, 164, 219, 285, 375, 483, 622, 799, 1017, 1284, 1621, 2033, 2537, 3158, 3915, 4832, 5953, 7303, 8930, 10896, 13248, 16071, 19451, 23482, 28272, 33977, 40736, 48741, 58201, 69367, 82506, 97986, 116139

Offset: 1

Author

Gus Wiseman, Apr 02 2023

Keywords

• nonn

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).

Examples

			The a(4) = 1 through a(9) = 15 partitions:
  (211)  (311)   (411)    (421)     (422)      (522)
         (2111)  (3111)   (511)     (521)      (621)
                 (21111)  (3211)    (611)      (711)
                          (4111)    (4211)     (4221)
                          (22111)   (5111)     (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (33111)
                                    (311111)   (42111)
                                    (2111111)  (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 >= 2*2, so y is counted under a(10).
		

Crossrefs

For length instead of median we have A237752.

For minimum instead of median we have A237821.

Reversing the inequality gives A361848.

The equal case is A361849, ranks A361856.

The unequal case is A361857, ranks A361867.

The complement is counted by A361858.

These partitions have ranks A361868.

For mean instead of median we have A361906.

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, A067538, A237755, A237820, A237824, A240219, A359907, A361851, A361860, A361907.

Programs

Mathematica

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