A363221 Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 19, 23, 26, 29, 37, 39, 49, 55, 62, 71, 84, 93, 108, 118, 141, 149, 188, 193, 217, 257, 279, 318, 369, 376, 441, 495, 572, 587, 692, 760, 811, 960, 1046, 1065, 1307, 1387, 1550, 1703, 1796, 2041, 2295, 2456, 2753, 3014

Offset

1,3

Comments

Also strict partitions such that (maximum) <= 2*(mean).

These are strict partitions whose complement (see A361851) has size <= n.

Links

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

Example

The partition y = (4,3,1) has length 3 and maximum 4, and 3*4 <= 2*8, so y is counted under a(8). The complement of y has size 4, which is less than or equal to n = 8.

Mathematica

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

Crossrefs

The equal case for median is A361850, non-strict A361849 (ranks A361856).

The non-strict version is A361851, A361848 for median.

The equal case is A361854, non-strict A361853 (ranks A361855).

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, A058398 by mean.

A051293 counts subsets with integer mean.

A067538 counts partitions with integer mean.

Cf. A111907, A237984, A240219, A241061, A241086, A324521, A324562, A349156, A360068, A360241, A361394, A361852, A361906.

Sequence in context: A238215 A237757 A027197 * A332577 A137793 A067659

Adjacent sequences: A363218 A363219 A363220 * A363222 A363223 A363224

Keyword

nonn

Author

Gus Wiseman, May 23 2023

Status

approved