A290517 -id:A290517 - cp's OEIS frontend

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

Alois P. Heinz, Aug 26 2019

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).

			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;
  ...

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

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.

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);

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 *)

A290518 Minimum value of Product_{i in lambda} i!, where lambda ranges over all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 2, 2, 6, 12, 12, 48, 144, 288, 288, 1440, 5760, 17280, 34560, 34560, 207360, 1036800, 4147200, 12441600, 24883200, 24883200, 174182400, 1045094400, 5225472000, 20901888000, 62705664000, 125411328000, 125411328000, 1003290624000, 7023034368000

Offset: 0

Alois P. Heinz, Aug 04 2017

nonn

			a(10) = 288 = (4! * 3! * 2! * 1!) is the value for partition [4,3,2,1]. All other partitions of 10 into distinct parts give larger values: [5,3,2]-> 1440, [5,4,1]-> 2880, [6,3,1]-> 4320, [6,4]-> 17280, [7,2,1]-> 10080, [7,3]-> 30240, [8,2]-> 80640, [9,1]-> 362880, [10]-> 3628800.

Alois P. Heinz, Table of n, a(n) for n = 0..960

Cf. A000142, A000217, A004736, A290517.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, infinity,
     `if`(n=0, 1, min(b(n, i-1), b(n-i, min(n-i, i-1))*i!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*
      (t-> t*(t+3)/2-n+2)(floor(sqrt(8*n-7)/2-1/2)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 05 2017

b[n_, i_]:=b[n, i]=If[n>i*(i + 1)/2, Infinity, If[n==0, 1, Min[b[n, i - 1], b[n - i, Min[n - i, i - 1]]*i!]]];  Table[b[n, n], {n, 0, 30}] (* Indranil Ghosh, Aug 05 2017, after Maple *)

a(n) = A000142(n) / A290517(n).
a(0) = 1, a(n) = a(n-1) * A004736(n) for n>0.
a(n) = a(n-1) iff n in { A000217 } \ { 0 }.

cp's OEIS Frontend

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.

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