A361851 Number of integer partitions of n such that (length) * (maximum) <= 2*n.

1, 2, 3, 5, 7, 11, 12, 18, 23, 31, 37, 51, 58, 75, 96, 116, 126, 184, 193, 253, 307, 346, 402, 511, 615, 678, 792, 1045, 1088, 1386, 1419, 1826, 2181, 2293, 2779, 3568, 3659, 3984, 4867, 5885, 6407, 7732, 8124, 9400, 11683, 13025, 13269, 16216, 17774, 22016

Offset

1,2

Comments

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

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

Links

Table of n, a(n) for n=1..50.

Example

The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (2221)
                                     (3111)    (3211)
                                     (21111)   (22111)
                                     (111111)  (211111)
                                               (1111111)
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 <= 2*7, so y is counted under a(7).
The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 is not <= 2*9, so y is not counted under a(9).
The partition y = (3,2,1,1) has diagram:
  o o o
  o o .
  o . .
  o . .
with complement of size 5, and 5 <= 7, so y is counted under a(7).

Mathematica

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

Crossrefs

For length instead of mean we have A237755.

For minimum instead of mean we have A237824.

For median instead of mean we have A361848.

The equal case for median is A361849, ranks A361856.

The unequal case is A361852, median A361858.

The equal case is A361853, ranks A361855.

Reversing the inequality gives 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.

Cf. A111907, A237984, A240219, A324521, A324562, A327482, A349156, A360068, A360071, A360241, A361394, A361859.

Sequence in context: A178357 A205667 A241506 * A372055 A165132 A193063

Adjacent sequences: A361848 A361849 A361850 * A361852 A361853 A361854

Keyword

nonn

Author

Gus Wiseman, Mar 28 2023

Status

approved