%I #6 Jan 21 2023 16:28:31

%S 0,1,1,2,2,3,4,4,4,7,6,6,10,7,10,13,11,9,20,10,20,18,21,12,30,24,28,

%T 27,30,15,73,16,37,43,45,67,74,19,55,71,126,21,150,22,75,225,78,24,

%U 183,126,245,192,132,27,284,244,403,303,120,30,828

%N Number of strict integer partitions of n whose parts have the same mean as median.

%e The a(1) = 1 through a(9) = 7 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8) (9)

%e (2,1) (3,1) (3,2) (4,2) (4,3) (5,3) (5,4)

%e (4,1) (5,1) (5,2) (6,2) (6,3)

%e (3,2,1) (6,1) (7,1) (7,2)

%e (8,1)

%e (4,3,2)

%e (5,3,1)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Mean[#]==Median[#]&]],{n,0,30}]

%Y The non-strict version is A240219, complement A359894, ranked by A359889.

%Y The complement is counted by A359898.

%Y The odd-length case is A359899, complement A359900.

%Y A000041 counts partitions, strict A000009.

%Y A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.

%Y A008289 counts strict partitions by mean.

%Y A237984 counts partitions containing their mean, complement A327472.

%Y A240850 counts strict partitions containing their mean, complement A240851.

%Y A325347 counts ptns with integer median, strict A359907, ranked by A359908.

%Y Cf. A065795, A066571, A067659, A082550, A102627, A135342, A316313, A327473, A327475, A328966, A359909.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 20 2023