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

0,2

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=0..49.

Formula

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

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

Example

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).

Mathematica

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

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.

Sequence in context: A283768 A212864 A026317 * A008768 A067593 A084993

Adjacent sequences: A361845 A361846 A361847 * A361849 A361850 A361851

Keyword

nonn

Author

Gus Wiseman, Mar 28 2023

Status

approved