A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

0, 0, 0, 1, 0, 1, 0, 3, 2, 2, 0, 7, 0, 3, 6, 10, 0, 13, 0, 17, 10, 5, 0, 40, 12, 6, 18, 34, 0, 62, 0, 50, 24, 8, 60, 125, 0, 9, 32, 169, 0, 165, 0, 95, 176, 11, 0, 373, 114, 198, 54, 143, 0, 384, 254, 574, 66, 14, 0, 1090, 0, 15, 748, 633, 448, 782, 0, 286

Offset: 1

Author

Gus Wiseman, Apr 10 2023

Keywords

• nonn

Comments

In terms of partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.

Examples

			The a(4) = 1 through a(12) = 7 partitions:
  (31)  .  (321)  .  (62)    (441)  (32221)  .  (93)
                     (3221)  (522)  (33211)     (642)
                     (3311)                     (4431)
                                                (5322)
                                                (322221)
                                                (332211)
                                                (333111)
The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:
  o o o o
  o o o o
  o o o .
  o . . .
Both the rectangle from the left and the complement have size 4.
		

Crossrefs

Positions of zeros are 1 and A000040.

For length instead of mean we have A237832.

For minimum instead of mean we have A118096.

These partitions have ranks A362047.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, A058398 by mean.

A067538 counts partitions with integer mean.

A097364 counts partitions by (maximum) - (minimum).

A243055 subtracts the least prime index from the greatest.

A326844 gives the diagram complement size of Heinz partition.

Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A349156, A359360, A360068, A360241, A361853.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Mean[#]&]],{n,30}]