A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 5, 0, 6, 3, 5, 0, 11, 6, 8, 7, 10, 0, 36, 0, 14, 16, 16, 29, 43, 0, 21, 36, 69, 0, 97, 0, 35, 138, 33, 0, 150, 61, 137, 134, 74, 0, 231, 134, 265, 229, 56, 0, 650, 0, 65, 749, 267, 247, 533, 0, 405, 565

Offset

1,12

Comments

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

These are strict partitions where both the diagram and its complement (see example) have size n.

Links

Table of n, a(n) for n=1..69.

Example

The a(n) strict partitions for selected n (A..E = 10..14):
  n=9:  n=12:  n=14:  n=15:  n=16:  n=18:  n=20:  n=21:  n=22:
--------------------------------------------------------------
  621   831    7421   A32    8431   C42    A532   E43    B542
        6321          A41    8521   C51    A541   E52    B632
                                    9432   A631   E61    B641
                                    9531   A721          B731
                                    9621   85421         B821
                                           86321
The a(20) = 6 strict partitions are: (10,7,2,1), (10,6,3,1), (10,5,4,1), (10,5,3,2), (8,6,3,2,1), (8,5,4,2,1).
The strict partition y = (8,5,4,2,1) has diagram:
  o o o o o o o o
  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(20).

Mathematica

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

Crossrefs

For minimum instead of mean we have A241035, non-strict A118096.

For length instead of mean we have A241087, non-strict A237753.

For median instead of mean we have A361850, non-strict A361849.

The non-strict version is A361853.

These partitions have ranks A361855 /\ A005117.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, A058398 by mean.

A008289 counts strict partitions by length.

A102627 counts strict partitions with integer mean, non-strict A067538.

A116608 counts partitions by number of distinct parts.

A268192 counts partitions by complement size, ranks A326844.

Cf. A111907, A237755, A240850, A326849 A359897, A360068, A360071, A360243, A361848, A361851, A361852, A361906.

Sequence in context: A160756 A306327 A176239 * A292475 A329903 A316716

Adjacent sequences: A361851 A361852 A361853 * A361855 A361856 A361857

Keyword

nonn

Author

Gus Wiseman, Mar 29 2023

Status

approved