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Number of integer partitions of n such that (length) = 2*(median).
4

%I #11 Apr 16 2023 06:35:13

%S 0,1,0,0,0,0,1,3,3,3,3,3,3,4,5,9,12,19,22,29,32,39,43,51,57,70,81,101,

%T 123,153,185,230,272,328,386,454,526,617,708,824,951,1106,1277,1493,

%U 1727,2020,2344,2733,3164,3684,4245,4914,5647,6502,7438,8533,9730

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

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). All of these partitions have even length, because an odd-length multiset cannot have fractional median.

%e The a(13) = 3 through a(15) = 5 partitions:

%e (7,2,2,2) (8,2,2,2) (9,2,2,2)

%e (8,2,2,1) (9,2,2,1) (10,2,2,1)

%e (8,3,1,1) (9,3,1,1) (10,3,1,1)

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

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

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

%Y For maximum instead of median we have A237753.

%Y For minimum instead of median we have A237757.

%Y For maximum instead of length we have A361849, ranks A361856.

%Y This is the equal case of A362048.

%Y These partitions have ranks A362050.

%Y A000041 counts integer partitions, strict A000009.

%Y A000975 counts subsets with integer median.

%Y A325347 counts partitions with integer median, complement A307683.

%Y A359893 and A359901 count partitions by median.

%Y A360005 gives twice median of prime indices, distinct A360457.

%Y Cf. A008284, A013580, A027193, A079309, A237800, A240219, A361801, A361848, A361858, A361859.

%K nonn

%O 1,8

%A _Gus Wiseman_, Apr 10 2023