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%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