different what?

[vdmcc]

Eric & Collin & others,
Formulating stuff
clearly is clearly
not my strongest point.
I meant to count 'multiplicities' as below:
Step_by_step:

Table[(Plus @@@ KSubsets[Range[2n], n]), {n, 0, 4}] // ColumnForm

  {0},
  {1, 2},
  {3, 4, 5, 5, 6, 7},
  {6, 7, 8, 9, 8, 9, 10, 10, 11, 12, 9, 10, 11, 11, 12, 13, 12, 13,
        14, 15},
  {10, 11, 12, 13, 14, 12, 13, 14, 15, 14, 15, 16, 16, 17, 18, 13,
        14, 15, 16, 15, 16, 17, 17, 18, 19, 16, 17, 18, 18, 19, 20, 19,
        20, 21, 22, 14, 15, 16, 17, 16, 17, 18, 18, 19, 20, 17, 18, 19,
        19, 20, 21, 20, 21, 22, 23, 18, 19, 20, 20, 21, 22, 21, 22, 23,
        24, 22, 23, 24, 25, 26}

**These are the outcomes (less offset):
Table[(Plus @@@ KSubsets[Range[2n], n]) - n(n + 1)/2, {n, 0, 4}]
{{0},
 {0, 1},
 {0, 1, 2, 2, 3, 4},
 {0, 1, 2, 3, 2, 3, 4, 4, 5, 6, 3, 4, 5, 5,
    6, 7, 6, 7, 8, 9},
 {0, 1, 2, 3, 4, 2, 3, 4, 5, 4, 5, 6, 6, 7, 8, 3, 4, 5,
    6, 5, 6, 7, 7, 8, 9, 6, 7, 8, 8, 9, 10, 9, 10, 11, 12, 4, 5, 6, 7, 6, 7,
    8, 8, 9, 10, 7, 8, 9, 9, 10, 11, 10, 11, 12, 13, 8, 9, 10, 10, 11, 12,
11,
     12, 13, 14, 12, 13, 14, 15, 16}}

** there are indeed n^2+1 different ones **
Table[Split[Sort[(Plus @@@ KSubsets[Range[2n], n]) - n(n + 1)/2]], {n, 0,
4}]

{{{0}},
 {{0}, {1}},
 {{0}, {1}, {2, 2}, {3}, {4}},
 {{0}, {1}, {2, 2}, {3, 3,
      3}, {4, 4, 4}, {5, 5, 5}, {6, 6, 6}, {7, 7}, {8}, {9}},
 {{0}, {1}, {2,
      2}, {3, 3, 3}, {4, 4, 4, 4, 4}, {5, 5, 5, 5, 5}, {6, 6, 6, 6, 6, 6,
      6}, {7, 7, 7, 7, 7, 7, 7}, {8, 8, 8, 8, 8, 8, 8, 8}, {9, 9, 9, 9, 9,
9,
      9}, {10, 10, 10, 10, 10, 10, 10}, {11, 11, 11, 11, 11}, {12, 12, 12,
12,
       12}, {13, 13, 13}, {14, 14}, {15}, {16}}}


** these are their frequencies, or multiplicities if you will **
** "how many KSubsets add up to a given value ? " **
Table[Length /@
    Split[Sort[(Plus @@@ KSubsets[Range[2n], n]) - n(n + 1)/2]], {n, 0, 4}]
{{1},
 {1, 1},
 {1, 1, 2, 1, 1},
 {1, 1, 2, 3, 3, 3, 3, 2, 1, 1},
 {1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1}}

** What I should have 'asked' is:
** "how many different multiplicities are there?"

Length[Union[#]] & /@
  Table[Length /@
      Split[Sort[(Plus @@@ KSubsets[Range[2n], n]) - n(n + 1)/2]], {n, 0,
4}]

{1, 1, 2, 3, 6, ..}

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