A309999 -id:A309999 - cp's OEIS frontend

A070289 Number of distinct values of multinomial coefficients ( n / (p1, p2, p3, ...) ) where (p1, p2, p3, ...) runs over all partitions of n.

1, 1, 2, 3, 5, 7, 11, 14, 20, 27, 36, 47, 64, 79, 102, 125, 157, 193, 243, 296, 366, 441, 538, 639, 773, 911, 1092, 1294, 1532, 1799, 2131, 2475, 2901, 3369, 3935, 4554, 5292, 6084, 7033, 8087, 9292, 10617, 12198, 13880, 15874, 18039, 20541, 23263, 26414, 29838

Offset: 0

Naohiro Nomoto, May 12 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..92
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, Journal of Number Theory 118 (2006), 15-30; arXiv preprint, arXiv:math/0509470 [math.CO], 2005.
Sergei Viznyuk, C-Program, C-Program, local copy.

Cf. A000041, A036038, A212855, A215520, A215521, A309951, A309999, A325305, A376661.

b:= proc(n,i) option remember;
      if n=0 then {1} elif i<1 then {} else {b(n, i-1)[],
         seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)} fi
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 14 2012

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Union[Join[b[n, i-1], Flatten[ Table[Function[{x}, x*i!^j] /@ b[n-i*j, i-1], {j, 1, n/i}]]]]]]; a[n_] := Length[b[n, n]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)

def A070289(n):
    P = Partitions(n)
    M = set(multinomial(list(x)) for x in P)
    return len(M)
[A070289(n) for n in range(20)]
# Joerg Arndt, Aug 14 2012

a(n) = A215520(n,n) = A215521(2*n,n). - Alois P. Heinz, Nov 08 2012

Terms a(n) for n >= 45 corrected by Joerg Arndt and Alois P. Heinz, Aug 14 2012

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.

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A070289 Number of distinct values of multinomial coefficients ( n / (p1, p2, p3, ...) ) where (p1, p2, p3, ...) runs over all partitions of n.

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