Sergei Viznyuk - cp's OEIS frontend

A213008 Triangle T(n,k) of number of distinct values of multinomial coefficients corresponding to sequence A026820 (n >= 1, 1 <= k <= n).

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 10, 12, 13, 14, 1, 5, 9, 14, 16, 18, 19, 20, 1, 5, 12, 17, 21, 23, 25, 26, 27, 1, 6, 13, 21, 26, 30, 32, 34, 35, 36, 1, 6, 16, 25, 33, 37, 41, 43, 45, 46, 47, 1, 7, 19, 32, 42, 50, 54, 58, 60, 62, 63, 64

Offset: 1

Sergei Viznyuk, Jun 01 2012

Differs from A026820 after position 24.

Includes sequence A070289 when k = n.

			Triangle T(n,k) begins:
  1;
  1, 2;
  1, 2, 3;
  1, 3, 4,  5;
  1, 3, 5,  6,  7;
  1, 4, 7,  9, 10, 11;
  1, 4, 8, 10, 12, 13, 14;
  ...
Thus, for n = 7 and k = 6 there are 13 distinct values of multinomial coefficients corresponding to partitions of n = 7 into at most k = 6 parts. The corresponding number of partitions from sequence A026820 is 14. That is because partitions 7 = 4 + 1 + 1 + 1 and 7 = 3 + 2 + 2 produce the same value of multinomial coefficient 7!/(4!*1!*1!*1!) = 7!/(3!*2!*2!).

Alois P. Heinz, Rows n = 1..45, flattened
Katsuhisa Yamanaka, Shin-ichiro Kawano, Yosuke Kikuchi, and Shin-ichi Nakano, Constant Time Generation of Integer Partitions, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E90-A, no.5, pp. 888-895, (May-2007).
Sergei Viznyuk, C-Program for this sequence, 2012.
Sergei Viznyuk, C-Program for sequences A026820, A070289, and A213008 (local copy), 2012.

Cf. A026820, A070289.

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

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1}, If[i<1, {}, Join[b[n, i-1, k], Table[ Function[#*i!^j] /@ b[n-i*j, i-1, k-j], {j, 1, Min[n/i, k]}] // Flatten] // Union] ]; T[n_, k_] := Length[b[n, n, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 12 2015, 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 *)

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

Original entry on oeis.org

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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A213008 Triangle T(n,k) of number of distinct values of multinomial coefficients corresponding to sequence A026820 (n >= 1, 1 <= k <= n).

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

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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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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.

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