A361859 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

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(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).

MATHEMATICA

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

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.

Sequence in context: A266813 A344232 A370639 * A192116 A088163 A048448

Adjacent sequences: A361856 A361857 A361858 * A361860 A361861 A361862

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 02 2023

STATUS

approved