A215520 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, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 10, 12, 13, 14, 1, 5, 10, 14, 17, 18, 19, 20, 1, 5, 12, 16, 21, 23, 25, 26, 27, 1, 6, 14, 20, 27, 29, 32, 34, 35, 36, 1, 6, 16, 22, 32, 35, 40, 43, 45, 46, 47, 1, 7, 19, 28, 40, 45, 52, 57, 60, 62, 63, 64

Offset: 1

Author

Alois P. Heinz, Aug 14 2012

Keywords

• nonn
• tabl

Comments

Differs from A026820 first at T(7,4).

Differs from A213008 first at T(8,3).

Examples

			T(3,2) =  2 = |{3!/(2!*1!), 3!/(1!*1!*1!)}| = |{3, 6}|.
T(5,2) =  3 = |{30, 60, 120}|.
T(7,4) = 10 = |{35, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040}|.
T(8,3) = 10 = |{560, 1120, 1680, 2520, 3360, 5040, 6720, 10080, 20160, 40320}|.
T(9,2) =  5 = |{22680, 45360, 90720, 181440, 362880}|.
Triangle T(n,k) begins:
  1;
  1,  2;
  1,  2,  3;
  1,  3,  4,  5;
  1,  3,  5,  6,  7;
  1,  4,  7,  9, 10, 11;
  1,  4,  8, 10, 12, 13, 14;
  1,  5, 10, 14, 17, 18, 19, 20;
  1,  5, 12, 16, 21, 23, 25, 26, 27;
  1,  6, 14, 20, 27, 29, 32, 34, 35, 36;
		

Links

• Alois P. Heinz, Rows n = 1..75, flattened
• Wikipedia, Multinomial coefficients

Crossrefs

Main diagonal gives: A070289.

Columns k=1-3 give: A000012, A008619 (for n>1), A001399 (for n>2).

Cf. A026820, A213008, A215521.

Programs

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)):
seq(seq(T(n, k), k=1..n), n=1..14);

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], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)