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Number of integer partitions of n such that the parts have the same mean as the multiplicities.
31

%I #15 Jul 09 2024 20:44:10

%S 1,1,0,0,1,0,0,0,2,1,0,0,6,0,0,0,6,0,7,0,1,0,0,0,0,90,0,63,0,0,0,0,11,

%T 0,0,0,436,0,0,0,0,0,0,0,0,2157,0,0,240,1595,22,0,0,0,6464,0,0,0,0,0,

%U 0,0,0,11628,4361,0,0,0,0,0,0,0,12927,0,0,621,0

%N Number of integer partitions of n such that the parts have the same mean as the multiplicities.

%C Note that such a partition cannot be strict for n > 1.

%C Conjecture: If n is squarefree, then a(n) = 0.

%e The n = 1, 4, 8, 9, 12, 16, 18 partitions (D=13):

%e (1) (22) (3311) (333) (322221) (4444) (444222)

%e (5111) (332211) (43222111) (444411)

%e (422211) (43321111) (552222)

%e (522111) (53221111) (555111)

%e (531111) (54211111) (771111)

%e (621111) (63211111) (822222)

%e (D11111)

%e For example, the partition (4,3,3,3,3,3,2,2,1,1) has mean 5/2, and its multiplicities (1,5,2,2) also have mean 5/2, so it is counted under a(20).

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

%Y These partitions are ranked by A359903, for prime factors A359904.

%Y Positions of positive terms are A360070.

%Y A000041 counts partitions, strict A000009.

%Y A058398 counts partitions by mean, see also A008284, A327482.

%Y A088529/A088530 gives mean of prime signature (A124010).

%Y A326567/A326568 gives mean of prime indices (A112798).

%Y A360069 counts partitions whose multiplicities have integer mean.

%Y Cf. A067340, A067538, A082550, A240219, A316313, A327475, A349156, A359893, A359897, A359905.

%K nonn

%O 0,9

%A _Gus Wiseman_, Jan 27 2023