A124773 -id:A124773 - cp's OEIS frontend

A124774 Multinomial coefficients for compositions in standard order.

1, 1, 1, 2, 1, 3, 3, 6, 1, 4, 6, 12, 4, 12, 12, 24, 1, 5, 10, 20, 10, 30, 30, 60, 5, 20, 30, 60, 20, 60, 60, 120, 1, 6, 15, 30, 20, 60, 60, 120, 15, 60, 90, 180, 60, 180, 180, 360, 6, 30, 60, 120, 60, 180, 180, 360, 30, 120, 180, 360, 120, 360, 360, 720, 1, 7, 21, 42, 35, 105

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

Number of ways to distribute labeled objects into boxes, with the number of objects in each box being specified by the composition.

			Composition number 11 is 2,1,1; there are 6 choices for the pair of objects in the first box, then 2 choices for the object in the next box, so a(11) = 6*2 = 12.
The table starts:
1
1
1 2
1 3 3 6

Alois P. Heinz, Rows n = 0..14, flattened
Thomas Garrity and Jacob Lehmann Duke, Ergodicity and Algebraticity of the Fast and Slow Triangle Maps, arXiv:2409.05822 [math.DS], 2024. See p. 22.

Cf. A066099, A124773, A011782 (row lengths), A000670 (row sums), A036039.

For composition b(1),...,b(k), a(n) = (Sum_{i=1}^k b(i))! / (Product_{i=1}^k b(i)!).

A124772 Number of set partitions associated with compositions in standard order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 3, 1, 2, 1, 1, 1, 4, 6, 6, 4, 8, 4, 4, 1, 3, 3, 3, 1, 2, 1, 1, 1, 5, 10, 10, 10, 20, 10, 10, 5, 15, 15, 15, 5, 10, 5, 5, 1, 4, 6, 6, 4, 8, 4, 4, 1, 3, 3, 3, 1, 2, 1, 1, 1, 6, 15, 15, 20, 40, 20, 20, 15, 45, 45, 45, 15, 30, 15, 15, 6, 24, 36, 36, 24, 48, 24, 24, 6, 18

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

Arrange the parts of the set partition by the smallest member of each part and read off the part sizes. E.g., for 1|24|3, the associated composition is 1,2,1. When the set partition is presented as the sequence of parts that each member is in, simply count the times each part number occurs. This representation for 1|24|3 is {1,2,3,2}.

			Composition number 11 is 2,1,1; the associated set partitions are 12|3|4, 13|2|4 and 14|2|3, so a(11) = 3.
The table starts:
1
1
1 1
1 2 1 1

Alois P. Heinz, Rows n = 0..14, flattened

Cf. A066099, A124773, A011782 (row lengths), A000110 (row sums), A036040, A080575.

For composition b(1),...,b(k), a(n) = Product_{i=1}^k C((Sum_{j=i}^k b(j))-1, b(i)-1).

A306549 a(n) is the product of the positions of the zeros in the binary expansion of n (the most significant bit having position 1).

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 3, 1, 24, 6, 8, 2, 12, 3, 4, 1, 120, 24, 30, 6, 40, 8, 10, 2, 60, 12, 15, 3, 20, 4, 5, 1, 720, 120, 144, 24, 180, 30, 36, 6, 240, 40, 48, 8, 60, 10, 12, 2, 360, 60, 72, 12, 90, 15, 18, 3, 120, 20, 24, 4, 30, 5, 6, 1, 5040, 720, 840, 120, 1008

Offset: 0

Rémy Sigrist, May 04 2019

Apparently, the variant where the least significant bit has position 1 corresponds to A124773.

			The first terms, alongside the positions of zeros and the binary representation of n, are:
  n   a(n)  Pos.zeros  bin(n)
  --  ----  ---------  ------
   0     1  {1}             0
   1     1  {}              1
   2     2  {2}            10
   3     1  {}             11
   4     6  {2,3}         100
   5     2  {2}           101
   6     3  {3}           110
   7     1  {}            111
   8    24  {2,3,4}      1000
   9     6  {2,3}        1001
  10     8  {2,4}        1010
  11     2  {2}          1011
  12    12  {3,4}        1100
  13     3  {3}          1101
  14     4  {4}          1110
  15     1  {}           1111

Rémy Sigrist, Table of n, a(n) for n = 0..16384

Cf. A070939, A124773, A306286.

A306549[n_] := Times @@ Flatten[Position[IntegerDigits[n, 2], 0]];
Array[A306549, 100, 0] (* Paolo Xausa, Jun 01 2024 *)

a(n) = my (b=binary(n)); prod(k=1, #b, if (b[k]==0, k, 1))

a(n) = vecprod(Vec(select(x->(x==0), binary(n), 1)));

from math import prod
def a(n): return prod(i for i, bi in enumerate(bin(n)[2:], 1) if bi == "0")
print([a(n) for n in range(70)]) # Michael S. Branicky, Jun 01 2024

a(n) = A070939(n)! / A306286(n).
a(2*n) = a(n) * (1+A070939(n)).
a(2*n+1) = a(n).
a(2^k) = (k+1)! for any k >= 0.
a(2^k-1) = 1 for any k >= 0.
a(2^k-2) = k for any k >= 1.

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A124774 Multinomial coefficients for compositions in standard order.

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A124772 Number of set partitions associated with compositions in standard order.

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A306549 a(n) is the product of the positions of the zeros in the binary expansion of n (the most significant bit having position 1).

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