A309992 Triangle T(n,k) whose n-th row lists in increasing order the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts; n >= 0, 1 <= k <= A000009(n), read by rows.

1, 1, 1, 1, 3, 1, 4, 1, 5, 10, 1, 6, 15, 60, 1, 7, 21, 35, 105, 1, 8, 28, 56, 168, 280, 1, 9, 36, 84, 126, 252, 504, 1260, 1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600, 1, 11, 55, 165, 330, 462, 495, 1320, 2310, 4620, 6930, 27720

Offset: 0

Author

Alois P. Heinz, Aug 26 2019

Keywords

• nonn
• tabf

Comments

First row with repeated terms is row 15, see also A309999: 1365 = M(15;11,4) = M(15;12,2,1) and 30030 = M(15;9,5,1) = M(15;10,3,2).

Examples

			For n = 5 there are 3 partitions of 5 into distinct parts: [5], [4,1], [3,2].  So row 5 contains M(5;5) = 1, M(5;4,1) = 5 and M(5;3,2) = 10.
Triangle T(n,k) begins:
  1;
  1;
  1;
  1,  3;
  1,  4;
  1,  5, 10;
  1,  6, 15,  60;
  1,  7, 21,  35, 105;
  1,  8, 28,  56, 168, 280;
  1,  9, 36,  84, 126, 252, 504, 1260;
  1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600;
  1, 11, 55, 165, 330, 462, 495, 1320, 2310,  4620, 6930, 27720;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..45, flattened
• Wikipedia, Multinomial coefficients
• Wikipedia, Partition (number theory)

Crossrefs

Columns k=1-3 give: A000012, A000027 (for n>=3), A000217(n-1) (for n>=5).

Row sums give A007837.

Rightmost terms of rows give A290517.

Cf. A000009, A036038, A309999, A325901, A325903.

Programs

Maple

g:= proc(n, i) option remember; `if`(i*(i+1)/2binomial(n, i)*x, g(n-i, min(n-i, i-1)))[], g(n, i-1)[]]))
    end:
T:= n-> sort(g(n$2))[]:
seq(T(n), n=0..14);

Mathematica

g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Join[ Binomial[n, i] # & /@ g[n-i, Min[n-i, i-1]], g[n, i-1]]]];
T[n_] := Sort[g[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)