A361907 Number of integer partitions of n such that (length) * (maximum) > 2*n.

0, 0, 0, 0, 0, 0, 3, 4, 7, 11, 19, 26, 43, 60, 80, 115, 171, 201, 297, 374, 485, 656, 853, 1064, 1343, 1758, 2218, 2673, 3477, 4218, 5423, 6523, 7962, 10017, 12104, 14409, 17978, 22031, 26318, 31453, 38176, 45442, 55137, 65775, 77451, 92533, 111485, 131057

Offset

1,7

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..48.

Example

The a(7) = 3 through a(10) = 11 partitions:
  (511)    (611)     (711)      (721)
  (4111)   (5111)    (5211)     (811)
  (31111)  (41111)   (6111)     (6211)
           (311111)  (42111)    (7111)
                     (51111)    (52111)
                     (411111)   (61111)
                     (3111111)  (421111)
                                (511111)
                                (3211111)
                                (4111111)
                                (31111111)
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 is not > 2*7, so y is not counted under a(7).
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 is not > 2*8, so y is not counted under a(8).
The partition y = (5,1,1,1) has length 4 and maximum 5, and 4*5 > 2*8, so y is counted under a(8).
The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 > 2*9, so y is counted under a(9).
The partition y = (3,2,1,1) has diagram:
  o o o
  o o .
  o . .
  o . .
with complement (shown in dots) of size 5, and 5 is not > 7, so y is not counted under a(7).

Mathematica

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

Crossrefs

For length instead of mean we have A237751, reverse A237754.

For minimum instead of mean we have A237820, reverse A053263.

The complement is counted by A361851, median A361848.

Reversing the inequality gives A361852.

The equal version is A361853.

For median instead of mean we have A361857, reverse A361858.

Allowing equality gives A361906, median A361859.

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, strict A102627, ranks A316413.

A116608 counts partitions by number of distinct parts.

A268192 counts partitions by complement size, ranks A326844.

Cf. A027193, A111907, A237752, A237755, A237821, A237824, A237984, A324562, A326622, A327482, A349156, A360071, A361394.

Sequence in context: A041739 A042593 A041018 * A072255 A049863 A025068

Adjacent sequences: A361904 A361905 A361906 * A361908 A361909 A361910

Keyword

nonn

Author

Gus Wiseman, Mar 29 2023

Status

approved