A085643 -id:A085643 - cp's OEIS frontend

A118851 Product of parts in n-th partition in Abramowitz and Stegun order.

1, 1, 2, 1, 3, 2, 1, 4, 3, 4, 2, 1, 5, 4, 6, 3, 4, 2, 1, 6, 5, 8, 9, 4, 6, 8, 3, 4, 2, 1, 7, 6, 10, 12, 5, 8, 9, 12, 4, 6, 8, 3, 4, 2, 1, 8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1, 9, 8, 14, 18, 20, 7, 12, 15, 16, 20, 24, 27, 6, 10, 12, 16, 18, 24, 5, 8, 9, 12, 16, 4

Offset: 0

Alford Arnold, May 01 2006

Let Theta(n) denote the set of norm values corresponding to all the partitions of n. The following results hold regarding this set: (i) Theta(n) is a subset of Theta(n+1); (ii) A prime p will appear as a norm only for partitions of n>=p; (iii) There exists a prime p not in Theta(n) for all n>=6; (iv) Let h(k) be the prime floor function which gives the greatest prime less than or equal to the k, then the prime p=h(n+1) does not belong to Theta(n); and (v) The primes not in the set Theta(n) are A000720(A000792(n)) - A000720(n). - Abhimanyu Kumar, Nov 25 2020

			a(9) = 4 because the 9th partition is [2,2] and 2*2 = 4.
Table T(n,k) starts:
  1;
  1;
  2, 1;
  3, 2,  1;
  4, 3,  4,  2,  1;
  5, 4,  6,  3,  4, 2,  1;
  6, 5,  8,  9,  4, 6,  8,  3,  4,  2, 1;
  7, 6, 10, 12,  5, 8,  9, 12,  4,  6, 8, 3, 4,  2,  1;
  8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1;

Abramowitz and Stegun, Handbook (1964) page 831.

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Walter Bridges and William Craig, On the distribution of the norm of partitions, arXiv:2308.00123 [math.CO], 2023.
Abhimanyu Kumar and Meenakshi Rana, On the treatment of partitions as factorization and further analysis, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).
Wolfdieter Lang, Rows n=1..10.
Robert Schneider and Andrew V. Sills, The Product of Parts or "Norm" of a Partition, INTEGERS, Volume 20A (2020) Paper #A13, 16 pp.
Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019-2021.

Cf. A000041 (row lengths), A006906 (row sums).

Cf. A005361, A036035, A036036, A048996, A078812, A085643, A103921, A111786.

C(sig)={vecprod(sig)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 19 2020

a(n) = A085643(n)/A048996(n).

T(n,k) = A005361(A036035(n,k)). - Andrew Howroyd, Oct 19 2020

Corrected and extended by Franklin T. Adams-Watters, May 26 2006

A305309 Array read by rows: a(n, k) = A048996(n, k) * A118851(n, k), n >= 1, k = 1..A000041(n).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 6, 4, 6, 1, 5, 8, 12, 9, 12, 8, 1, 6, 10, 16, 9, 12, 36, 8, 12, 24, 10, 1, 7, 12, 20, 24, 15, 48, 27, 36, 16, 72, 32, 15, 40, 12, 1, 8, 14, 24, 30, 16, 18, 60, 72, 48, 54, 20, 96, 54, 144, 16, 20, 120, 80, 18, 60, 14, 1, 9, 16, 28, 36, 40, 21, 72, 90, 48, 60, 144, 27, 24, 120, 144, 192, 216, 96, 25, 160, 90, 360, 80, 24, 180, 160, 21, 84, 16, 1, 10, 18, 32, 42, 48, 25, 24, 84, 108, 120, 72, 180, 96, 108, 28, 144, 180, 96, 240, 576, 108, 128, 216, 30, 200, 240, 480, 540, 480, 32, 30, 240, 135, 720, 240, 28, 252, 280, 24, 112, 18, 1

Offset: 1

Wolfdieter Lang, May 31 2018

The Data section here is longer than usual.  Do not shorten it! - N. J. A. Sloane, Jan 10 2019
The length of row n is A000041(n), the number of partitions of n.
Partitions follow the Abramowitz-Stegun (A-St) order (see the link).
The row sums give A001906(n) = Fibonacci(2*n).
The triangle T(n, m) obtained by summing in row n the entries of the columns k with identical part number m is A078812(n, m) = binomial(n+m-1, 2*m-1) (with offsets n >= 1, m = 1..n). The array of the number of parts m = m(n,k) = A036043(n, k) in A-St order.
This array is the elementwise product of the array A048996, the composition numbers, and A118851, the products of the parts of partitions, both arrays are in A-St order.
Therefore a(n, k) is the sum of the number of products of the block lengths of all the A048996(n, k) set partition of [n] := {1,2, ..., n} with m = m(n, k) blocks consisting of consecutive numbers corresponding to the k-th partition of n in A-St order. Because the block structure depends only on the exponents (signature) of the underlying partition this leads to the product of the two array entries. Equivalently, one can consider compositions. Then a(n, k) gives the sum of the products of the parts of all A048996(n, k) compositions originating from the k-th partition of n.
This array is the result of an attempt to understand the comment of Kevin Long, May 11 2018, on A001906.
This array is similar to A085643 but some pairs of numbers like  (27, 36), (72,48), (54,144), ... are there swapped.

			For the rows n = 1..10, and comments on compositions and set partitions with blocks of consecutive numbers, see the link.
Example: n = 5, k = 4: the partition is (1^2, 3^1) = [1,1,3] with m = m(n,k) = 3. The A048996(5, 4) = 3 compositions are 1 + 1 + 3, 1 + 3 + 1 and 3 + 1 + 1. The corresponding three consecutive 3-block partitions of [5] := {1, 2, ..., 5} are {1}, {2}, {3,4,5} and {1}, {2,3,4}, {5} and {1,2,3}, {4}, {5}, Therefore, a(5, 4) = 1*1*3 + 1*3*1 + 3*1*1 = 3*3 = 9. For the compositions one has the same sum from the products of the parts.

Milton Abramowitz and Irene A. Stegun, editors, Multinomials and Partitions, Handbook of Mathematical Functions, December 1972, pp. 831-2.
Wolfdieter Lang, Rows n = 1..10, and more.

Cf. A000041, A001906, A036043, A048996, A078812, A085643, A118851.

a(n, k) = A048996(n, k) * A118851(n, k), n >= 1, k = 1..A000041(n).

A115980 Array read by rows distributing the values of A000712 (vertically) and A001519 (horizontally).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 3, 2, 4, 6, 5, 4, 6, 2, 4, 6, 8, 6, 3, 4, 4, 8, 9, 2, 4, 6, 8, 10, 7, 4, 12, 8, 4, 8, 12, 12, 2, 4, 6, 8, 10, 12, 8, 3, 6, 5, 4, 14, 21, 12, 4, 8, 12, 16, 15, 2, 4, 6, 8, 10, 12, 14, 9

Offset: 0

Alford Arnold, Feb 11 2006

A001906 records the partial sums of the column sequence A001519 and is also the row sum of A078812 and of A085643; sequences linking a(n) to compositions of n having k parts when there are q kinds of part q. - Alford Arnold, Apr 30 2006

			The array begins:
1
..2
....2
....3
......2
......4..3
......4
.........2
.........4..4..3
.........6..6..4
.........5
with column sums beginning 1 2 5 10 20 ...A000712 related to A000041
and sums over each template beginning 1 2 5 13 34 ...A001519 related to A000045

Cf. A000041, A000045.

Cf. A001906, A078812, A085643.

cp's OEIS Frontend

A118851 Product of parts in n-th partition in Abramowitz and Stegun order.

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A305309 Array read by rows: a(n, k) = A048996(n, k) * A118851(n, k), n >= 1, k = 1..A000041(n).

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A115980 Array read by rows distributing the values of A000712 (vertically) and A001519 (horizontally).

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