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Number of strict integer partitions of n with non-integer mean.
2

%I #11 Mar 16 2023 12:23:44

%S 1,0,0,1,0,2,0,4,2,4,5,11,0,17,15,13,15,37,18,53,24,48,78,103,23,111,

%T 152,143,123,255,110,339,238,372,495,377,243,759,845,873,414,1259,842,

%U 1609,1383,1225,2281,2589,1285,2827,2518,3904,3836,5119,3715,4630

%N Number of strict integer partitions of n with non-integer mean.

%C Are 1, 2, 4, 6, 12 the only zeros?

%e The a(3) = 1 through a(11) = 11 partitions:

%e {2,1} . {3,2} . {4,3} {4,3,1} {5,4} {5,3,2} {6,5}

%e {4,1} {5,2} {5,2,1} {6,3} {5,4,1} {7,4}

%e {6,1} {7,2} {6,3,1} {8,3}

%e {4,2,1} {8,1} {7,2,1} {9,2}

%e {4,3,2,1} {10,1}

%e {5,4,2}

%e {6,3,2}

%e {6,4,1}

%e {7,3,1}

%e {8,2,1}

%e {5,3,2,1}

%p a:= proc(m) option remember; local b; b:=

%p proc(n, i, t) option remember; `if`(i*(i+1)/2<n,

%p 0, `if`(n=0, signum(irem(m, t)),

%p b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))

%p end: `if`(m=0, 1, b(m$2, 0))

%p end:

%p seq(a(n), n=0..55); # _Alois P. Heinz_, Mar 16 2023

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

%Y The strict complement is counted by A102627.

%Y The non-strict version is ranked by A348551, complement A316413.

%Y The non-strict version is counted by A349156, complement A067538.

%Y For median instead of mean we have A360952, complement A359907.

%Y A000041 counts partitions, strict A000009.

%Y A008284/A058398/A327482 count partitions by mean.

%Y A307683 counts partitions with non-integer median, ranks A359912.

%Y A325347 counts partitions with integer median, ranks A359908.

%Y A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.

%Y A327472 counts partitions not containing their mean, complement of A237984.

%Y A327475 counts subsets with integer mean.

%Y Cf. A051293, A082550, A143773, A175397, A175761, A240219, A240850, A326027, A326641, A326849, A359897.

%K nonn

%O 0,6

%A _Gus Wiseman_, Mar 11 2023

%E a(31)-a(55) from _Alois P. Heinz_, Mar 16 2023