A376367 -id:A376367 - 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.

A376376 Least number that can be written as a multinomial coefficient in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

2, 10, 6, 420, 120, 210, 4324320, 7207200, 720720, 360360, 6983776800, 9777287520, 13967553600, 48886437600, 195545750400, 24736537425600, 586637251200, 293318625600, 148419224553600, 742096122768000, 28941748787952000, 296838449107200, 1736504927277120000

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

a(n) is the least number that occurs exactly n times in A036038 or A376367, i.e., the least number m such that A376369(m) = n.
After a(62), the sequence continues (where "?" represents terms that are either 0 or greater than 10^29): ?, 92098021748598694855458432000, ?, 6268725246643132945351680000, 1567181311660783236337920000, ?, 3134362623321566472675840000. After a(69), all terms are either 0 or larger than 10^29.
It seems that a(n) often is in A025487, at least for small n. The exceptions are n = 2, 12, 26, 30, 31, 33, 34, 35, 36, 37, 38, 42, 44, ... .

Pontus von Brömssen, Table of n, a(n) for n = 1..62

First column of A376370.

Cf. A025487, A036038, A376367, A376369.

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.

A376373 Numbers that occur exactly 3 times 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 3 integer partitions (x_1, ..., x_k).

Original entry on oeis.org

6, 20, 30, 56, 60, 90, 105, 252, 360, 462, 495, 504, 560, 720, 756, 990, 1320, 1365, 1540, 1716, 2970, 3360, 3960, 4290, 4620, 5460, 6006, 6435, 7920, 8190, 10080, 10296, 10626, 10920, 11628, 12012, 12870, 14280, 15504, 17550, 18360, 21840, 23256, 24024, 24310

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 3, i.e., numbers that appear exactly 3 times in A376367.

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

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

Third row of A376370.
Subsequence of A325593.
Cf. A036038, A376367, A376369.

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.

A376374 Numbers that occur exactly 4 times 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 4 integer partitions (x_1, ..., x_k).

Original entry on oeis.org

420, 630, 840, 1980, 3003, 7140, 7560, 9240, 13860, 15120, 25200, 43680, 53130, 55440, 72072, 90090, 116280, 120120, 142506, 277200, 278256, 332640, 371280, 415800, 450450, 480480, 813960, 1113840, 1261260, 1801800, 2018940, 2441880, 2702700, 3255840, 3326400

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 4, i.e., numbers that appear exactly 4 times in A376367.

			420 is a term, because it can be represented as a multinomial coefficient in exactly 4 ways: 420 = 420!/(1!*419!) = 21!/(1!*1!*19!) = 8!/(2!*2!*4!) = 7!/(1!*1!*2!*3!).

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

Fourth row of A376370.

Cf. A036038, A376367, A376369.

A376375 Numbers that occur exactly 5 times 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 5 integer partitions (x_1, ..., x_k).

Original entry on oeis.org

120, 1680, 60060, 83160, 180180, 240240, 831600, 900900, 1081080, 1627920, 1663200, 2522520, 2882880, 3603600, 7567560, 10090080, 14414400, 20180160, 25225200, 30270240, 35814240, 36756720, 37837800, 46558512, 49008960, 51482970, 60540480, 61261200, 64864800

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 5, i.e., numbers that appear exactly 5 times in A376367.

			120 is a term, because it can be represented as a multinomial coefficient in exactly 5 ways: 120 = 120!/(1!*119!) = 16!/(2!*14!) = 10!/(3!*7!) = 6!/(1!*1!*1!*3!) = 5!/(1!*1!*1!*1!*1).

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

Fifth row of A376370.

Cf. A036038, A376367, A376369.

A376379 Heinz numbers of integer partitions (x_1, ..., x_k) with at least 2 parts, sorted by increasing multinomial coefficients (x_1 + ... + x_k)!/(x_1! * ... * x_k!). In case of ties, the partitions are sorted in standard order as in A080577.

Original entry on oeis.org

4, 6, 10, 14, 8, 9, 22, 26, 34, 38, 15, 46, 58, 12, 62, 74, 82, 21, 86, 94, 106, 118, 122, 20, 25, 134, 33, 142, 146, 158, 16, 166, 178, 194, 202, 39, 206, 214, 18, 28, 218, 226, 254, 262, 274, 35, 278, 51, 298, 302, 314, 326, 334, 346, 44, 358, 362, 382, 57

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

This is a permutation of the composite numbers A002808.

			  n | A376367(n) | partition | a(n)
  --+------------+-----------+-----
  1 |     2      |  (1,1)    |   4
  2 |     3      |  (2,1)    |   6
  3 |     4      |  (3,1)    |  10
  4 |     5      |  (4,1)    |  14
  5 |     6      |  (1,1,1)  |   8
  6 |     6      |  (2,2)    |   9
  7 |     6      |  (5,1)    |  22
The number 210 appears 6 times in A376367, corresponding to the partitions (4,1,1,1), (3,2,2), (6,4), (13,1,1), (19,2), and (209,1), with Heinz numbers 56, 45, 91, 164, 201 and 2578, respectively. These numbers appear as a(257), ..., a(262).

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

Cf. A002808, A080577, A318762, A376367.

A318762(a(n)) = A376367(n).

A376840 Take the integer partitions with at least 2 parts in order of their associated multinomial coefficients; a(n) is the sum of the n-th partition, i.e., the number of the row of A036038 (or A078760) in which the multinomial coefficient appears. In case of ties, take the sums (or row numbers) in nondecreasing order.

Original entry on oeis.org

2, 3, 4, 5, 3, 4, 6, 7, 8, 9, 5, 10, 11, 4, 12, 13, 14, 6, 15, 16, 17, 18, 19, 5, 6, 20, 7, 21, 22, 23, 4, 24, 25, 26, 27, 8, 28, 29, 5, 6, 30, 31, 32, 33, 34, 7, 35, 9, 36, 37, 38, 39, 40, 41, 7, 42, 43, 44, 10, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 11, 55

Offset: 1

Pontus von Brömssen, Oct 06 2024

nonn

Equivalently, a(n) is the number of the row of A036038 (or A078760) in which A376367(n) appears, with row numbers in nondecreasing order for numbers that appear multiple times in A376367.

The multinomial coefficient of the n-th partition, with the ordering considered here, is A376367(n).

			  n | A376367(n) | partition | a(n)
  --+------------+-----------+-----
  1 |     2      |  (1,1)    |  2
  2 |     3      |  (2,1)    |  3
  3 |     4      |  (3,1)    |  4
  4 |     5      |  (4,1)    |  5
  5 |     6      |  (1,1,1)  |  3
  6 |     6      |  (2,2)    |  4
  7 |     6      |  (5,1)    |  6

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

Cf. A036038, A056239, A078760, A376367, A376379.

a(n) = A056239(A376379(n)).

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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A376376 Least number that can be written as a multinomial coefficient in exactly n ways, or 0 if no such number exists.

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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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A376373 Numbers that occur exactly 3 times 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 3 integer partitions (x_1, ..., x_k).

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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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A376374 Numbers that occur exactly 4 times 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 4 integer partitions (x_1, ..., x_k).

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A376375 Numbers that occur exactly 5 times 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 5 integer partitions (x_1, ..., x_k).

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A376379 Heinz numbers of integer partitions (x_1, ..., x_k) with at least 2 parts, sorted by increasing multinomial coefficients (x_1 + ... + x_k)!/(x_1! * ... * x_k!). In case of ties, the partitions are sorted in standard order as in A080577.

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