A361853 - OEIS

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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

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OFFSET

1,6

COMMENTS

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

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

LINKS

Table of n, a(n) for n=1..62.

EXAMPLE

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 . . . . .

Since the partition and its complement (shown in dots) have the same size, y is counted under a(12).

MATHEMATICA

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

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.

Sequence in context: A117316 A109189 A264157 * A144172 A227318 A166692

Adjacent sequences: A361850 A361851 A361852 * A361854 A361855 A361856

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 29 2023

STATUS

approved