A210238 -id:A210238 - 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 *)

A209936 Triangle of multiplicities of k-th partition of n corresponding to sequence A080577. Multiplicity of a given partition of n into k parts is the number of ways parts can be selected from k distinguishable bins. See the example.

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, 60, 15, 120, 60, 20, 90, 30, 1, 7, 42, 42, 105, 42, 210, 140, 105, 105, 420, 105, 140, 210, 42, 1, 8, 56, 56, 168, 56, 336, 280, 28, 336, 168, 840, 280, 168, 420, 840, 1120, 168, 70, 560, 420, 56, 1

Offset: 1

Sergei Viznyuk, Mar 15 2012

Differs from A035206 after position 21.
Differs from A210238 after position 21.
The n-th row of the triangle, written as a column vector v(n), satisfies  K . v(n) = #SSYT(lambda,n) where K is the Kostka matrix of order n, and #SSYT(lambda,n) is the count of semi-standard Young tableaux in n variables of the partitions of n. - Wouter Meeussen, Jan 27 2025

			Triangle begins:
  1
  2, 1
  3, 6, 1
  4, 12, 6, 12, 1
  5, 20, 20, 30, 30, 20, 1
  6, 30, 30, 60, 15, 120, 60, 20, 90, 30, 1
  7, 42, 42, 105, 42, 210, 140, 105, 105, 420, 105, 140, 210, 42, 1
  ...
Thus for n=3 (third row) the partitions of n=3 are:
  3+0+0  0+3+0  0+0+3   (multiplicity=3),
  2+1+0  2+0+1  1+2+0  1+0+2  0+2+1  0+1+2  (multiplicity=6),
  1+1+1  (multiplicity=1).

Sergei Viznyuk, C Program

Cf. A080577, A078760, A035206, A210238.
Row lengths give A000041.
Row sums give A088218.

Apply[Multinomial,Last/@Tally[#]&/@PadRight[IntegerPartitions[n]],1] (* Wouter Meeussen, Jan 26 2025 *)

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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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