A361391 Number of strict integer partitions of n with non-integer mean.

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, 152, 143, 123, 255, 110, 339, 238, 372, 495, 377, 243, 759, 845, 873, 414, 1259, 842, 1609, 1383, 1225, 2281, 2589, 1285, 2827, 2518, 3904, 3836, 5119, 3715, 4630

Offset

0,6

Comments

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

Links

Table of n, a(n) for n=0..55.

Example

The a(3) = 1 through a(11) = 11 partitions:
  {2,1}  .  {3,2}  .  {4,3}    {4,3,1}  {5,4}  {5,3,2}    {6,5}
            {4,1}     {5,2}    {5,2,1}  {6,3}  {5,4,1}    {7,4}
                      {6,1}             {7,2}  {6,3,1}    {8,3}
                      {4,2,1}           {8,1}  {7,2,1}    {9,2}
                                               {4,3,2,1}  {10,1}
                                                          {5,4,2}
                                                          {6,3,2}
                                                          {6,4,1}
                                                          {7,3,1}
                                                          {8,2,1}
                                                          {5,3,2,1}

Maple

a:= proc(m) option remember; local b; b:=
      proc(n, i, t) option remember; `if`(i*(i+1)/2<n,
        0, `if`(n=0, signum(irem(m, t)),
        b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
      end: `if`(m=0, 1, b(m$2, 0))
    end:
seq(a(n), n=0..55);  # Alois P. Heinz, Mar 16 2023

Mathematica

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

Crossrefs

The strict complement is counted by A102627.

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

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

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

A000041 counts partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

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

A325347 counts partitions with integer median, ranks A359908.

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

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

A327475 counts subsets with integer mean.

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

Sequence in context: A332001 A196606 A371130 * A337697 A328599 A222303

Adjacent sequences: A361388 A361389 A361390 * A361392 A361393 A361394

Keyword

nonn

Author

Gus Wiseman, Mar 11 2023

Extensions

a(31)-a(55) from Alois P. Heinz, Mar 16 2023

Status

approved