A209936 -id:A209936 - cp's OEIS frontend

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

Sergei Viznyuk, Mar 18 2012

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.

			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

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

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

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

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 *)

A210238 Triangle of multiplicities D(n) of multinomial coefficients corresponding to sequence A210237.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1, 8, 56, 56, 224, 28, 336, 336, 280, 168, 168, 840, 420, 1120, 70, 1120, 560, 168, 420, 56, 1

Offset: 1

Sergei Viznyuk, Mar 18 2012

Multiplicity D(n) of multinomial coefficient M(n) is the number of ways the same value of M(n)=n!/(m1!*m2!*..*mk!) is obtained by distributing n identical balls into k distinguishable bins.
Differs from A209936 after a(21).
Differs from A035206 after a(36).
The checksum relationship: sum(M(n)*D(n)) = k^n
The number of terms per row (for each value of n starting with n=1) forms sequence A070289.

			1
2, 1
3, 6, 1
4, 12, 6, 12, 1
5, 20, 20, 30, 30, 20, 1
6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1
7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1
Thus for n=3 (third row) the same value of multinomial coefficient follows from the following combinations:
3!/(3!0!0!) 3!/(0!3!0!) 3!/(0!0!3!) (i.e. multiplicity=3)
3!/(2!1!0!) 3!/(2!0!1!) 3!/(0!2!1!) 3!/(0!1!2!) 3!/(1!0!2!) 3!/(1!2!0!)  (i.e. multiplicity=6)
3!/(1!1!1!) (i.e. multiplicity=1)

Sergei Viznyuk, C-program for the sequence.

Cf. A210237, A209936, A035206, A070289.

Table[Last/@Tally[Multinomial@@@Compositions[k,k]],{k,8}] (* Wouter Meeussen, Mar 09 2013 *)

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

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