A210237 Triangle of distinct values M(n) of multinomial coefficients for partitions of n in increasing order of n and M(n).

1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 20, 30, 60, 90, 120, 180, 360, 720, 1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040, 1, 8, 28, 56, 70, 168, 280, 336, 420, 560, 840, 1120, 1680, 2520, 3360, 5040, 6720

Offset

1,3

Comments

Differs from A036038 after a(37). To illustrate where the difference comes from, consider 4,1,1,1 and 3,2,2 are two different partitions of 7 having the same value of multinomial coefficient M(n)=n!/(m1!*m2!*...*mk!)=210.

There is no known formula for M(n) sequence, however the asymptotic behavior has been studied, see the paper by Andrews, Knopfmacher, and Zimmermann.

The number of terms per row (for each value of n starting with n=1) forms sequence A070289.

Links

Alois P. Heinz, Rows n = 1..29, flattened

George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the Number of Distinct Multinomial Coefficients, arXiv:math/0509470 [math.CO], 2005.

Sergei Viznyuk, C-program for the sequence

Example

Trianglebegins:
  1;
  1, 2;
  1, 3,  6;
  1, 4,  6, 12, 24;
  1, 5, 10, 20, 30,  60, 120;
  1, 6, 15, 20, 30,  60,  90, 120, 180, 360, 720;
  1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040;
  ...
Thus for n=4 (fourth row) the distinct values of multinomial coefficients are:
  4!/(4!) = 1
  4!/(3!1!) = 4
  4!/(2!2!) = 6
  4!/(2!1!1!) = 12
  4!/(1!1!1!1!) = 24

Maple

b:= proc(n, i) option remember; `if`(n=0 or i<2, {1},
      {seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=0..n/i)})
    end:
T:= n-> sort([map(x-> n!/x, b(n, n))[]])[]:
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 13 2012

Mathematica

b[n_, i_] := b[n, i] = If[n==0 || i<2, {1}, Union[Flatten @ Table[(#*i!^j&) /@ b[n-i*j, i-1], {j, 0, n/i}]]]; T[n_] := Sort[Flatten[n!/#& /@ b[n, n]] ]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

Crossrefs

Cf. A036038, A210238, A078760, A209936, A080577, A070289.

Sequence in context: A338797 A171999 A036038 * A078760 A348113 A103280

Adjacent sequences: A210234 A210235 A210236 * A210238 A210239 A210240

Keyword

nonn,tabf

Author

Sergei Viznyuk, Mar 18 2012

Status

approved