A078760 -id:A078760 - cp's OEIS frontend

A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Pontus von Brömssen, Sep 22 2024

nonn

a(n) is the number of occurrences of n in each of A036038, A050382, A078760, A318762, and A376367.

The sequence is unbounded. To see this, note that the sets of parts (1,1,1,4) and (2,2,3) of a partition can be exchanged without affecting the value of the multinomial coefficient, because 1+1+1+4 = 2+2+3 and 1!*1!*1!*4! = 2!*2!*3!. In particular, a((7*k)!/24^k) >= k+1 from the partitions 7*k = (3*j)*1 + j*4 + (2*(k-j))*2 + (k-j)*3 for 0 <= j <= k.

			a(6) = 3, because 6 can be written as a multinomial coefficient in 3 ways: 6 = 6!/(1!*5!) = 4!/(2!*2!) = 3!/(1!*1!*1!).

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A036038, A050382, A078760, A318762, A376367, A376368, A376370.

A376367 Sorted multinomial coefficients greater than 1, including duplicates.

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 20, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 29, 30, 30, 30, 31, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53

Offset: 1

Pontus von Brömssen, Sep 22 2024

nonn

Sorted terms of A036038, A050382, A078760, or A318762, excluding 1 (which appears infinitely often).

The number k appears A376369(k) times.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A036038, A050382, A078760, A318762, A376369, A376379.

a(n) = A318762(A376379(n)).

A376661 Frequency of the most common number among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 6, 6, 7, 8, 9, 11, 11, 13, 13, 14, 15, 16, 18, 19, 20, 23, 24, 26, 27, 30, 33, 37, 40, 43, 49, 52, 57, 64, 68, 76, 79, 87, 93, 99, 109, 116, 125, 135, 143, 157, 171, 191, 206, 223, 238, 254, 276, 291

Offset: 0

Pontus von Brömssen, Oct 02 2024

nonn

Frequency of the most common number in row n of A036038 (for n >= 1) or A078760.
The sequence is nondecreasing, because a set of partitions of n-1 with a common multinomial coefficient can be extended to a set of partitions of n with a common multinomial coefficient by adding a unit part to each partition. It appears that a(n) > a(n-1) for n >= 28.
The sequence is unbounded. To see this, note that the sets of parts (1,1,1,4) and (2,2,3) of a partition can be exchanged without affecting the value of the multinomial coefficient, because 1+1+1+4 = 2+2+3 and 1!*1!*1!*4! = 2!*2!*3!. In particular, a((7*k)!/24^k) >= k+1 from the partitions 7*k = (3*j)*1 + j*4 + (2*(k-j))*2 + (k-j)*3 for 0 <= j <= k.

			For n = 7, the only number that appears more than once in row 7 of A036038 is 210, which appears twice: 210 = 7!/(2!*2!*3!) = 7!/(1!*1!*1!*4!). Hence, a(7) = 2.

Cf. A036038, A070289, A078760, A376369, A376662, A376663.

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 *)

A309972 Product of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n.

Original entry on oeis.org

1, 1, 2, 18, 6912, 216000000, 1632586752000000000, 498266101635303733401600000000000, 1140494258799407218656986754465090350453096448000000000000000

Offset: 0

Alois P. Heinz, Aug 25 2019

nonn

			a(3) = M(3;3) * M(3;2,1) * M(3;1,1,1) = 1 * 3 * 6 = 18.

Alois P. Heinz, Table of n, a(n) for n = 0..14
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Rightmost terms in rows of A309951.

Cf. A000041, A005651, A036038, A078760, A210237.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [n!], [map(t->
      binomial(n, i)*t, b(n-i, min(n-i, i)))[], b(n, i-1)[]])
    end:
a:= n-> mul(i, i=b(n$2)):
seq(a(n), n=0..9);  # Alois P. Heinz, Aug 25 2019

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {n!}, Join[Binomial[n, i] #& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
a[n_] := Times @@ b[n, n];
a /@ Range[0, 9] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

a(n) = Product_{k=1..A000041(n)} A036038(n,k).

a(n) = A309951(n,A000041(n)).

A376368 Least number k with a partition k = x_1 + ... + x_j such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n.

Original entry on oeis.org

0, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 14, 6, 16, 17, 18, 19, 5, 7, 22, 23, 4, 25, 26, 27, 8, 29, 5, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 7, 43, 44, 10, 46, 47, 48, 49, 50, 51, 52, 53, 54, 11, 8, 57, 58, 59, 5, 61, 62, 63, 64, 65, 12, 67, 68, 69, 8, 71

Offset: 1

Pontus von Brömssen, Sep 22 2024

nonn

Index of first row of A078760 (or A036038 when n >= 2) that contains n.

a(n) <= n, with equality if and only if n is in A376371, i.e., if and only if n is not in A325472.

			a(6) = 3, because 6 appears in row 3 of A078760, corresponding to the multinomial coefficient 3!/(1!*1!*1!) = 6.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Pontus von Brömssen, Log-log plot, using Plot2.

Cf. A036038, A078760, A325472, A376369, A376370, A376371.

a(k!) = k for k != 1.

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 *)

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
  n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
m
1         1
2         2      1
3         6      3     1
4        24     12     4     1
5       120     60    20     5    1
6       720    360   120    30    6   1
7      5040   2520   840   210   42   7   1
8     40320  20160  6720  1680  336  56   8   1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Tilman Piesk, Triangle rows m=0..25, flattened
Tilman Piesk, Illustration of the multisets for m=0..6

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A376662 The smallest of the most common numbers among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 210, 56, 504, 1260, 9240, 166320, 360360, 5045040, 75675600, 1210809600, 4084080, 73513440, 698377680, 13967553600, 146659312800, 1075501627200, 37104806138400, 296838449107200, 7420961227680000, 96472495959840000, 2604757390915680000

Offset: 0

Pontus von Brömssen, Oct 02 2024

nonn

a(n) is the smallest number that appears A376661(n) times in row n of A036038 (for n >= 1) or A078760.

			For n = 8, the only numbers that appear more than once in row 8 of A036038 are 56 and 1680, which both appear twice. Since 56 < 1680, a(8) = 56.

Cf. A036038, A078760, A376661.

cp's OEIS Frontend

A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

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A376367 Sorted multinomial coefficients greater than 1, including duplicates.

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A376661 Frequency of the most common number among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

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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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Maple

Mathematica

A309972 Product of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n.

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Maple

Mathematica

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A376368 Least number k with a partition k = x_1 + ... + x_j such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to 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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Mathematica

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

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A376662 The smallest of the most common numbers among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

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