A325472 -id:A325472 - cp's OEIS frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

2, 3, 10, 4, 12, 6, 5, 15, 20, 420, 7, 21, 30, 630, 120, 8, 24, 56, 840, 1680, 210, 9, 28, 60, 1980, 60060, 1260, 4324320, 11, 35, 90, 3003, 83160, 2520, 21621600, 7207200, 13, 36, 105, 7140, 180180, 5040, 24504480, 151351200, 720720

Offset: 1

Pontus von Brömssen, Sep 22 2024

Row n lists numbers m such that A376369(m) = n.
In case there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.
Any integer k >= 2 appears exactly once in the array.

			Array begins:
  n\k|       1         2         3         4         5          6          7          8
  ---+---------------------------------------------------------------------------------
  1  |       2         3         4         5         7          8          9         11
  2  |      10        12        15        21        24         28         35         36
  3  |       6        20        30        56        60         90        105        252
  4  |     420       630       840      1980      3003       7140       7560       9240
  5  |     120      1680     60060     83160    180180     240240     831600     900900
  6  |     210      1260      2520      5040     27720     166320    1441440    4084080
  7  | 4324320  21621600  24504480  43243200  75675600  116396280  367567200  908107200
  8  | 7207200 151351200 302702400 411863760 823727520 1816214400 2327925600 4655851200

Pontus von Brömssen, Table of n, a(n) for n = 1..1081 (antidiagonals 1..46)

Cf. A036038, A050382, A078760, A318762, A325472 (complement of first row), A325593 (complement of the union of the first 2 rows), A376369, A376376 (first column).

First five rows are A376371, A376372, A376373, A376374, A376375.

A325306 Numbers which are represented by more than one partition of the same integer.

Original entry on oeis.org

56, 210, 504, 1260, 1365, 1680, 1716, 2520, 5040, 7560, 9240, 13860, 15120, 17550, 21840, 24024, 25200, 25740, 27720, 30030, 42504, 43680, 55440, 60060, 69300, 72072, 75600, 77520, 83160, 110880, 120120, 151200, 154440, 166320, 168168, 180180, 185640, 203490

Offset: 1

Peter Luschny, Sep 06 2019

nonn

We call (p1+p2+ ...)! / (p1!*p2!*p3! ...) a 'partition coefficient' of n if (p1, p2, p3, ...) is a partition and n = p1 + p2 + ... .

We say 'n is represented by p' if n is the partition coefficient of p.

			56 is in this list because it is represented by [5, 3] and [6, 1, 1].
210 is in this list because it is represented by [3, 2, 2] and [4, 1, 1, 1].
These are 'irreducible pairs' of partitions in the terminology of Andrews et al.
Note that the terms can derive from different integers. For instance 27720 is represented by [6, 2, 1, 1, 1] and [5, 3, 2, 1] (partitions of 11) and also by [6, 4, 1, 1] and [5, 4, 3] (partitions of 12).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the Number of Distinct Multinomial Coefficients, arXiv:math/0509470 [math.CO], 2005.

Cf. A036038, A309897, A325472.

from collections import Counter
def A325306_list(n):
    res = []
    for k in range(2*n):
        L = A309897(k)
        d = Counter(L)
        res += [j for j, v in d.items() if v > 1]
    return sorted(Set(res))[:n]
A325306_list(20)

A325593 Numbers having at least three representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.

Original entry on oeis.org

1, 6, 20, 30, 56, 60, 90, 105, 120, 210, 252, 360, 420, 462, 495, 504, 560, 630, 720, 756, 840, 990, 1260, 1320, 1365, 1540, 1680, 1716, 1980, 2520, 2970, 3003, 3360, 3960, 4290, 4620, 5040, 5460, 6006, 6435, 7140, 7560, 7920, 8190, 9240, 10080, 10296, 10626

Offset: 1

Alois P. Heinz, Sep 06 2019

nonn

Numbers occurring at least three times in the triangle A036038.

			1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
6 is in the sequence because M(6;5,1) = M(4;2,2) = M(3;1,1,1) = 6.
56 is in the sequence because M(56;55,1) = M(8;5,3) = M(8;6,1,1) = 56.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000 (first 505 terms from Alois P. Heinz)

Cf. A036038, A325472, A325903, A376370.

A376373 is a subsequence.

A325901 Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n into distinct parts.

Original entry on oeis.org

1, 10, 15, 21, 28, 35, 36, 45, 55, 56, 60, 66, 78, 84, 91, 105, 120, 126, 136, 153, 165, 168, 171, 190, 210, 220, 231, 252, 253, 276, 280, 286, 300, 325, 330, 351, 360, 364, 378, 406, 435, 455, 462, 465, 495, 496, 504, 528, 560, 561, 595, 630, 660, 666, 680

Offset: 1

Alois P. Heinz, Sep 07 2019

nonn

Numbers that are repeated in the triangle A309992 (all positive integers except 2 occur at least once).
All triangular numbers (A000217) except 0, 3 and 6 are in this sequence.
All terms are also contained in A325472.

			1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
10 is in the sequence because M(10;9,1) = M(5;3,2) = 10.
55 is in the sequence because M(55;54,1) = M(11;9,2) = 55.
105 is in the sequence because M(7;4,2,1) = M(15;13,2) = M(105;104,1) = 105.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A000009, A000217, A309992, A325472, A325903.

A376371 Numbers that occur exactly once in A036038, i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!), with 1 <= x_1 <= ... <= x_k, is equal to m only when (x_1, ..., x_k) = (1, m-1).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 18, 19, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 1, i.e., numbers that appear only once in A376367.

			10 is not a term, because it can be represented as a multinomial coefficient in 2 ways: 10 = 10!/(1!*9!) = 5!/(2!*3!).

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

First row of A376370.
Complement of A325472 (with respect to the positive integers).
Cf. A036038, A376367, A376369.

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.

A376372 Numbers that occur exactly twice in A036038, i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly 2 integer partitions (x_1, ..., x_k).

Original entry on oeis.org

10, 12, 15, 21, 24, 28, 35, 36, 42, 45, 55, 66, 70, 72, 78, 84, 91, 110, 126, 132, 136, 140, 153, 156, 165, 168, 171, 180, 182, 190, 220, 231, 240, 253, 272, 276, 280, 286, 300, 306, 325, 330, 336, 342, 351, 364, 378, 380, 406, 435, 455, 465, 496, 506, 528, 552

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 2, i.e., numbers that appear exactly twice in A376367.

			10 is a term, because it can be represented as a multinomial coefficient in exactly 2 ways: 10 = 10!/(1!*9!) = 5!/(2!*3!).

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

Second row of A376370.
Subsequence of A325472.
Cf. A036038, A376367, A376369.

A376664 Least number k such that there are A376663(n) partitions x_1 + ... + x_j = k such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n, i.e., the first row k of A036038 in which n appears A376663(n) times (or 0 if n = 0).

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, Oct 02 2024

nonn

Differs from A376368 at n = 1260, 2520, 5040, 60060, 83160, ... . For example, 1260 appears first in row A376368(1260) = 7 of A036038, but only once. It also appears once in row 9, but in row a(1260) = 10 it appears A376663(1260) = 2 times.

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

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

Cf. A036038, A325472, A376368, A376663.

cp's OEIS Frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

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A325306 Numbers which are represented by more than one partition of the same integer.

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SageMath

A325593 Numbers having at least three representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.

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A325901 Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n into distinct parts.

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A376371 Numbers that occur exactly once in A036038, i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!), with 1 <= x_1 <= ... <= x_k, is equal to m only when (x_1, ..., x_k) = (1, m-1).

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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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Formula

A376372 Numbers that occur exactly twice in A036038, i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly 2 integer partitions (x_1, ..., x_k).

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A376664 Least number k such that there are A376663(n) partitions x_1 + ... + x_j = k such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n, i.e., the first row k of A036038 in which n appears A376663(n) times (or 0 if n = 0).

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