%I #7 Mar 31 2023 05:01:14

%S 0,0,0,0,0,0,0,0,1,0,0,2,0,1,2,2,0,5,0,6,3,5,0,11,6,8,7,10,0,36,0,14,

%T 16,16,29,43,0,21,36,69,0,97,0,35,138,33,0,150,61,137,134,74,0,231,

%U 134,265,229,56,0,650,0,65,749,267,247,533,0,405,565

%N Number of strict integer partitions of n such that (length) * (maximum) = 2n.

%C Also strict partitions satisfying (maximum) = 2*(mean).

%C These are strict partitions where both the diagram and its complement (see example) have size n.

%e The a(n) strict partitions for selected n (A..E = 10..14):

%e n=9: n=12: n=14: n=15: n=16: n=18: n=20: n=21: n=22:

%e --------------------------------------------------------------

%e 621 831 7421 A32 8431 C42 A532 E43 B542

%e 6321 A41 8521 C51 A541 E52 B632

%e 9432 A631 E61 B641

%e 9531 A721 B731

%e 9621 85421 B821

%e 86321

%e The a(20) = 6 strict partitions are: (10,7,2,1), (10,6,3,1), (10,5,4,1), (10,5,3,2), (8,6,3,2,1), (8,5,4,2,1).

%e The strict partition y = (8,5,4,2,1) has diagram:

%e o o o o o o o o

%e o o o o o . . .

%e o o o o . . . .

%e o o . . . . . .

%e o . . . . . . .

%e Since the partition and its complement (shown in dots) have the same size, y is counted under a(20).

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

%Y For minimum instead of mean we have A241035, non-strict A118096.

%Y For length instead of mean we have A241087, non-strict A237753.

%Y For median instead of mean we have A361850, non-strict A361849.

%Y The non-strict version is A361853.

%Y These partitions have ranks A361855 /\ A005117.

%Y A000041 counts integer partitions, strict A000009.

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

%Y A008289 counts strict partitions by length.

%Y A102627 counts strict partitions with integer mean, non-strict A067538.

%Y A116608 counts partitions by number of distinct parts.

%Y A268192 counts partitions by complement size, ranks A326844.

%Y Cf. A111907, A237755, A240850, A326849 A359897, A360068, A360071, A360243, A361848, A361851, A361852, A361906.

%K nonn

%O 1,12

%A _Gus Wiseman_, Mar 29 2023