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Number of strict odd-length integer partitions of n whose parts do not have the same mean as median.
12

%I #7 Jan 21 2023 09:33:25

%S 0,0,0,0,0,0,0,1,2,1,4,5,4,8,10,8,15,18,17,26,27,31,43,51,53,59,81,87,

%T 109,127,115,169,194,213,255,243,322,379,431,478,487,629,667,804,907,

%U 902,1151,1294,1439,1530,1674,2031,2290,2559,2829,2973,3296,3939

%N Number of strict odd-length integer partitions of n whose parts do not have the same mean as median.

%e The a(7) = 1 through a(16) = 15 partitions (A=10, B=11, C=12, D=13):

%e (421) (431) (621) (532) (542) (651) (643) (653) (762) (754)

%e (521) (541) (632) (732) (652) (743) (843) (763)

%e (631) (641) (831) (742) (752) (861) (853)

%e (721) (731) (921) (751) (761) (942) (862)

%e (821) (832) (842) (A32) (871)

%e (841) (851) (A41) (943)

%e (931) (932) (B31) (952)

%e (A21) (941) (C21) (961)

%e (A31) (A42)

%e (B21) (A51)

%e (B32)

%e (B41)

%e (C31)

%e (D21)

%e (64321)

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

%Y This is the strict case of A359896, complement A359895, ranked by A359892.

%Y This is the odd-length case of A359898, complement A359897.

%Y The complement is counted by A359899.

%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 A027193 counts odd-length partitions, strict A067659, ranked by A026424.

%Y A359893/A359901/A359902 count partitions by median, ranked by A360005.

%Y Cf. A000016, A065795, A066571, A102627, A240850, A240851, A327475, A359894, A359906, A359907, A359910.

%K nonn

%O 0,9

%A _Gus Wiseman_, Jan 21 2023