A325306 -id:A325306 - cp's OEIS frontend

A325472 Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.

1, 6, 10, 12, 15, 20, 21, 24, 28, 30, 35, 36, 42, 45, 55, 56, 60, 66, 70, 72, 78, 84, 90, 91, 105, 110, 120, 126, 132, 136, 140, 153, 156, 165, 168, 171, 180, 182, 190, 210, 220, 231, 240, 252, 253, 272, 276, 280, 286, 300, 306, 325, 330, 336, 342, 351, 360

Offset: 1

Alois P. Heinz, Sep 06 2019

nonn

Numbers that are repeated in the triangle A036038 (all positive integers occur at least once).

All triangular numbers (A000217) except 0 and 3 are in this sequence.

			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.
42 is in the sequence because M(42;41,1) = M(7;5,1,1) = 42.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A000041, A000217, A036038, A305188, A325306, A325593, A325901.

a(n) = A305188(n-1) for n > 1.

A376667 Square array read by antidiagonals: row n lists numbers whose maximal frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., numbers m such that A376663(m) = n.

Original entry on oeis.org

1, 2, 56, 3, 210, 166320, 4, 504, 360360, 4084080, 5, 1260, 720720, 17907120, 1396755360, 6, 1365, 2162160, 73513440, 4190266080, 698377680, 7, 1680, 5045040, 75675600, 4655851200, 13967553600, 146659312800, 8, 1716, 5765760, 220540320, 4942365120, 27935107200, 293318625600, 1075501627200

Offset: 1

Pontus von Brömssen, Oct 02 2024

In case there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

Each positive integer appears exactly once in the array, so as a linear sequence it is a permutation of the positive integers (unless there are any 0's).

			Array begins:
  n\k|             1             2              3              4              5              6
  ---+----------------------------------------------------------------------------------------
  1  |             1             2              3              4              5              6
  2  |            56           210            504           1260           1365           1680
  3  |        166320        360360         720720        2162160        5045040        5765760
  4  |       4084080      17907120       73513440       75675600      220540320      411863760
  5  |    1396755360    4190266080     4655851200     4942365120     9884730240    24443218800
  6  |     698377680   13967553600    27935107200   267711444000   537750813600   586637251200
  7  |  146659312800  293318625600  1606268664000  3226504881600  6184134356400  7228208988000
  8  | 1075501627200 6453009763200 12368268712800 24736537425600 29683844910720 74209612276800

Pontus von Brömssen, Table of n, a(n) for n = 1..465 (antidiagonals 1..30)

Cf. A036038, A078760, A325306 (complement of first row), A376370, A376663, A376673 (first column).

First five rows are A376668, A376669, A376670, A376671, A376672.

A376668 Positive integers that do not appear more than once in the same row of A036038 (or A078760), i.e., numbers m such that A376663(m) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

Is this the same as A357759? - R. J. Mathar, Oct 09 2024. [Answer: No, they are different. - Andrew Howroyd, Oct 09 2024]

			56 is not a term, because it can be represented as a multinomial coefficient for 2 different partitions of 8: 56 = 8!/(1!*1!*6!) = 8!/(3!*5!).

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

First row of A376667.
Complement of A325306 (with respect to the positive integers).
Cf. A036038, A078760, A376371, A376663.

A376669 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 2, i.e., numbers m such that A376663(m) = 2.

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, 168168, 180180, 185640, 203490, 240240

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			56 is a term, because it can be represented as a multinomial coefficient for 2 different partitions of 8 (and never for more than 2 different partitions of the same integer): 56 = 8!/(1!*1!*6!) = 8!/(3!*5!).

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

Second row of A376667.
Subsequence of A325306.
Cf. A036038, A078760, A376372, A376663.

A376821 Number of irreducible pairs of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 3, 3, 6, 7, 8, 4, 6, 7, 12, 17, 23, 23, 31, 38, 36, 70, 71, 101, 127, 118, 145, 191, 209, 261, 309, 396, 462, 512, 652, 769, 878, 1097, 1320, 1563, 1827, 2098, 2533, 2932, 3475, 4185, 4756, 5726, 6614, 7686, 9189, 10825

Offset: 0

Pontus von Brömssen, Oct 05 2024

nonn

A pair of partitions of n is irreducible if the two partitions yield the same multinomial coefficient but have no parts in common. The partitions in the pair are required to be distinct, otherwise a(0) would be 1.

			   n | irreducible pairs of partitions of n
  ---+-------------------------------------
   7 | (1,1,1,4), (2,2,3)
   8 | (1,1,6), (3,5)
  10 | (1,4,5), (2,2,6)
  13 | (1,1,1,10), (6,7)
     | (1,1,3,8), (2,4,7)
     | (1,1,1,1,1,8), (2,2,2,7)
  14 | (1,2,2,9), (3,3,8)
     | (1,1,1,2,9), (3,4,7)
     | (1,1,1,1,1,1,4,4), (2,2,2,2,3,3)

George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, Journal of Number Theory 118 (2006), 15-30; arXiv preprint, arXiv:math/0509470 [math.CO], 2005. (See Section 7.)

Cf. A070289, A260669, A325306, A376661.

A327410 Numbers represented by the partition coefficients of prime partitions.

Original entry on oeis.org

1, 6, 10, 20, 21, 36, 56, 78, 90, 105, 120, 171, 210, 252, 300, 364, 465, 528, 560, 741, 756, 792, 903, 990, 1140, 1176, 1485, 1540, 1680, 1830, 1953, 1980, 2346, 2520, 2600, 2628, 2775, 3240, 3432, 3570, 4095, 4368, 4851, 4960, 5253, 5460, 5886, 5984, 6105

Offset: 1

Peter Luschny, Sep 07 2019

nonn

Given a partition pi = (p1, p2, p3, ...) we call the associated multinomial coefficient (p1+p2+ ...)! / (p1!*p2!*p3! ...) the 'partition coefficient' of pi and denote it by . We say 'k is represented by pi' if k = .

A partition is a prime partition if all parts are prime.

			(2*n)!/2^n (for n >= 1) is a subsequence because [2,2,...,2] (n times '2') is a prime partition. Similarly A327411(n) is a subsequence because [3,2,2,...,2] (n times '2') is a prime partition. (3*n)!/(6^n) and A327412 are subsequences for the same reason.
The representations are not unique. 1 is the represented by all partitions of the form [p], p prime. For example 210 is represented by [3, 2, 2] and by [19, 2]. The list below shows the partitions with the smallest sum.
1   <- [2],
6   <- [2, 2],
10  <- [3, 2],
20  <- [3, 3],
21  <- [5, 2],
36  <- [7, 2],
56  <- [5, 3],
78  <- [11, 2],
90  <- [2, 2, 2],
105 <- [13, 2],
120 <- [7, 3],
171 <- [17, 2],
210 <- [3, 2, 2],
252 <- [5, 5],
300 <- [23, 2].

George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, arXiv:math/0509470 [math.CO], 2005.
Eric Weisstein's World of Mathematics, Prime Partition

Cf. A000607, A036038, A325306, A000680, A327411, A014606, A327412.

def A327410_list(n):
    res = []
    for k in range(2*n):
        P = Partitions(k, parts_in = prime_range(k+1))
        res += [multinomial(p) for p in P]
    return sorted(Set(res))[:n]
print(A327410_list(20))

cp's OEIS Frontend

A325472 Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.

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A376667 Square array read by antidiagonals: row n lists numbers whose maximal frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., numbers m such that A376663(m) = n.

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A376668 Positive integers that do not appear more than once in the same row of A036038 (or A078760), i.e., numbers m such that A376663(m) = 1.

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A376669 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 2, i.e., numbers m such that A376663(m) = 2.

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A376821 Number of irreducible pairs of partitions of n.

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A327410 Numbers represented by the partition coefficients of prime partitions.

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SageMath