A361848 Number of integer partitions of n such that (maximum) <= 2*(median).

1, 2, 3, 5, 6, 9, 12, 15, 19, 26, 31, 40, 49, 61, 75, 93, 112, 137, 165, 199, 238, 289, 341, 408, 482, 571, 674, 796, 932, 1096, 1280, 1495, 1738, 2026, 2347, 2724, 3148, 3639, 4191, 4831, 5545, 6372, 7298, 8358, 9552, 10915, 12439, 14176, 16121, 18325

Offset: 0

Author

Gus Wiseman, Mar 28 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(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (321)     (331)
                                     (2211)    (421)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (22111)
                                               (211111)
                                               (1111111)
For example, the partition y = (3,2,2) has maximum 3 and median 2, and 3 <= 2*2, so y is counted under a(7).
		

Crossrefs

For length instead of median we have A237755.

For minimum instead of median we have A237824.

The equal case is A361849, ranks A361856.

For mean instead of median we have A361851.

The complement is counted by A361857, ranks A361867.

The unequal case is A361858.

Reversing the inequality gives 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, A013580, A027193, A061395, A067538, A111907, A240219, A324562, A359907, A361394, A361860.

Programs

Mathematica

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

Formula

a(n) = A361849(n) + A361858(n).

a(n) = A000041(n) - A361857(n).