A362049 Number of integer partitions of n such that (length) = 2*(median).

0, 1, 0, 0, 0, 0, 1, 3, 3, 3, 3, 3, 3, 4, 5, 9, 12, 19, 22, 29, 32, 39, 43, 51, 57, 70, 81, 101, 123, 153, 185, 230, 272, 328, 386, 454, 526, 617, 708, 824, 951, 1106, 1277, 1493, 1727, 2020, 2344, 2733, 3164, 3684, 4245, 4914, 5647, 6502, 7438, 8533, 9730

Offset

1,8

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). All of these partitions have even length, because an odd-length multiset cannot have fractional median.

Links

Table of n, a(n) for n=1..57.

Example

The a(13) = 3 through a(15) = 5 partitions:
  (7,2,2,2)  (8,2,2,2)      (9,2,2,2)
  (8,2,2,1)  (9,2,2,1)      (10,2,2,1)
  (8,3,1,1)  (9,3,1,1)      (10,3,1,1)
             (3,3,3,3,1,1)  (3,3,3,3,2,1)
                            (4,3,3,3,1,1)

Mathematica

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

Crossrefs

For maximum instead of median we have A237753.

For minimum instead of median we have A237757.

For maximum instead of length we have A361849, ranks A361856.

This is the equal case of A362048.

These partitions have ranks A362050.

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, A079309, A237800, A240219, A361801, A361848, A361858, A361859.

Sequence in context: A081168 A301415 A210509 * A333819 A216944 A178832

Adjacent sequences: A362046 A362047 A362048 * A362050 A362051 A362052

Keyword

nonn

Author

Gus Wiseman, Apr 10 2023

Status

approved