A361906 Number of integer partitions of n such that (length) * (maximum) >= 2*n.

0, 0, 0, 0, 0, 2, 3, 5, 9, 15, 19, 36, 43, 68, 96, 125, 171, 232, 297, 418, 529, 676, 853, 1156, 1393, 1786, 2316, 2827, 3477, 4484, 5423, 6677, 8156, 10065, 12538, 15121, 17978, 22091, 26666, 32363, 38176, 46640, 55137, 66895, 79589, 92621, 111485, 133485

Offset: 1

Author

Gus Wiseman, Mar 29 2023

Keywords

• nonn

Comments

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

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

Examples

			The a(6) = 2 through a(10) = 15 partitions:
  (411)   (511)    (611)     (621)      (721)
  (3111)  (4111)   (4211)    (711)      (811)
          (31111)  (5111)    (5211)     (5221)
                   (41111)   (6111)     (5311)
                   (311111)  (42111)    (6211)
                             (51111)    (7111)
                             (321111)   (42211)
                             (411111)   (43111)
                             (3111111)  (52111)
                                        (61111)
                                        (421111)
                                        (511111)
                                        (3211111)
                                        (4111111)
                                        (31111111)
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 >= 2*8, so y is counted under a(8).
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 = (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).
		

Crossrefs

For length instead of mean we have A237752, reverse A237755.

For minimum instead of mean we have A237821, reverse A237824.

For median instead of mean we have A361859, reverse A361848.

The unequal case is A361907.

The complement is counted by A361852.

The equal case is A361853, ranks A361855.

Reversing the inequality gives A361851.

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.

A268192 counts partitions by complement size, ranks A326844.

Cf. A027193, A111907, A116608, A237984, A324521, A327482, A349156, A360068.

Programs

Mathematica

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