A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

0, 0, 0, 0, 0, 2, 0, 1, 2, 4, 0, 10, 0, 8, 16, 10, 0, 31, 0, 44, 44, 20, 0, 92, 50, 28, 98, 154, 0, 266, 0, 154, 194, 48, 434, 712, 0, 60, 348, 910, 0, 1198, 0, 1120, 2138, 88, 0, 2428, 1300, 1680, 912, 2506, 0, 4808, 4800, 5968, 1372, 140, 0, 14820, 0, 160

Offset: 1

Author

Gus Wiseman, Mar 29 2023

Keywords

• nonn

Comments

Also partitions satisfying (maximum) = 2*(mean).

These are partitions whose diagram has the same size as its complement (see example).

Examples

			The a(6) = 2 through a(12) = 10 partitions:
  (411)   .  (4211)  (621)     (5221)   .  (822)
  (3111)             (321111)  (5311)      (831)
                               (42211)     (6222)
                               (43111)     (6321)
                                           (6411)
                                           (422211)
                                           (432111)
                                           (441111)
                                           (32211111)
                                           (33111111)
The partition y = (6,4,1,1) has diagram:
  o o o o o o
  o o o o . .
  o . . . . .
  o . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(12).
		

Crossrefs

For minimum instead of mean we have A118096.

For length instead of mean we have A237753.

For median instead of mean we have A361849, ranks A361856.

This is the equal case of A361851, unequal case A361852.

The strict case is A361854.

These partitions have ranks A361855.

This is the equal case of A361906, unequal case A361907.

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.

A268192 counts partitions by complement size, ranks A326844.

Cf. A111907, A116608, A188814, A237755, A237824, A237984, A240219, A326849, A327482, A349156, A359894.

Programs

Mathematica

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