A215521 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part = k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 1, 5, 10, 10, 10, 7, 5, 3, 2, 1, 1, 1, 6, 12, 14, 12, 11, 7, 5, 3, 2, 1, 1, 1, 6, 14, 16, 17, 13, 11, 7, 5, 3, 2, 1, 1

Offset

1,8

Comments

Differs from A008284 first at T(11,4).

Links

Alois P. Heinz, Rows n = 1..100, flattened

Wikipedia, Multinomial coefficients

Example

T(4,2) = 2 = |{4!/(2!*2!), 4!/(2!*1!*1!)}| = |{6, 12}|.
T(7,4) = 3 = |{35, 105, 210}|.
T(8,3) = 5 = |{560, 1120, 1680, 3360, 6720}|.
T(11,4) = 10 = |{11550, 34650, 46200, 69300, 138600, 207900, 277200, 415800, 831600, 1663200}|.
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  1,  1;
  1,  2,  2,  1,  1;
  1,  3,  3,  2,  1,  1;
  1,  3,  4,  3,  2,  1,  1;
  1,  4,  5,  5,  3,  2,  1,  1;
  1,  4,  7,  6,  5,  3,  2,  1,  1;
  1,  5,  8,  9,  7,  5,  3,  2,  1,  1;
  1,  5, 10, 10, 10,  7,  5,  3,  2,  1,  1;
  ...

Maple

b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
T:= (n, k)-> nops(b(n-k, min(k, n-k))):
seq(seq(T(n, k), k=1..n), n=1..15);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Join[b[n, i - 1], Table[ b[n - i*j, i - 1] *i!^j, {j, 1, n/i}] // Flatten]] // Union]; T[n_, k_] := Length[b[n, k]]; Table[Table[T[n - k, Min[k, n - k]], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)

Crossrefs

Columns k=1-3 give: A000012 (for n>0), A004526, A069905(n) = A001399(n-3) for n>=3.

T(2*n,n) gives: A070289.

Cf. A008284, A215520.

Sequence in context: A166240 A219347 A114087 * A008284 A114088 A208245

Adjacent sequences: A215518 A215519 A215520 * A215522 A215523 A215524

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 14 2012

Status

approved