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

Author

Gus Wiseman, May 23 2023

Keywords

• nonn

Comments

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

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

Examples

			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.
		

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.

Programs

Mathematica

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