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

0,5

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).

Links

Alois P. Heinz, Rows n = 0..45, flattened

Wikipedia, Multinomial coefficients

Wikipedia, Partition (number theory)

Example

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;
  ...

Maple

g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, [], `if`(n=0, [1],
     [map(x->binomial(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 *)

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.

Sequence in context: A187079 A086183 A014462 * A016474 A332678 A069264

Adjacent sequences: A309989 A309990 A309991 * A309993 A309994 A309995

Keyword

nonn,tabf

Author

Alois P. Heinz, Aug 26 2019

Status

approved