%I #9 Apr 12 2023 11:08:26

%S 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,

%T 0,50,24,8,60,125,0,9,32,169,0,165,0,95,176,11,0,373,114,198,54,143,0,

%U 384,254,574,66,14,0,1090,0,15,748,633,448,782,0,286

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

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

%e The a(4) = 1 through a(12) = 7 partitions:

%e (31) . (321) . (62) (441) (32221) . (93)

%e (3221) (522) (33211) (642)

%e (3311) (4431)

%e (5322)

%e (322221)

%e (332211)

%e (333111)

%e 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:

%e o o o o

%e o o o o

%e o o o .

%e o . . .

%e Both the rectangle from the left and the complement have size 4.

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

%Y Positions of zeros are 1 and A000040.

%Y For length instead of mean we have A237832.

%Y For minimum instead of mean we have A118096.

%Y These partitions have ranks A362047.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, A058398 by mean.

%Y A067538 counts partitions with integer mean.

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

%Y A243055 subtracts the least prime index from the greatest.

%Y A326844 gives the diagram complement size of Heinz partition.

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

%K nonn

%O 1,8

%A _Gus Wiseman_, Apr 10 2023