A212721 Triangle read by rows: n-th row gives distinct products of partitions of n (A000041).

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

Offset: 0

Author

Reinhard Zumkeller, Jun 14 2012

Keywords

• nonn
• tabf
• look

Comments

A034891(n) = length of n-th row;

A000792(n) = largest term of n-th row;

for n>5: A007918(n) = smallest number <= A000792(n) not occurring in n-th row.

Examples

			A000041(6)=11, the 11 partitions and their products of 6:
   1: (1,1,1,1,1,1)   ->   1 * 1 * 1 * 1 * 1 * 1 = 1
   2: (1,1,1,1,2)     ->   1 * 1 * 1 * 1 * 2     = 2
   3: (1,1,1,3)       ->   1 * 1 * 1 * 3         = 3
   4: (1,1,2,2)       ->   1 * 1 * 2 * 2         = 4
   5: (1,1,4)         ->   1 * 1 * 4             = 4
   6: (1,2,3)         ->   1 * 2 * 3             = 6
   7: (1,5)           ->   1 * 5                 = 5
   8: (2,2,2)         ->   2 * 2 * 2             = 8
   9: (2,4)           ->   2 * 4                 = 8
  10: (3,3)           ->   3 * 3                 = 9
  11: (6)             ->                           6,
sorted and duplicates removed: T(6,1..8)=[1,2,3,4,5,6,8,9], A034891(6)=8.
The triangle begins:
   0 |  [1]
   1 |  [1]
   2 |  [1,2]
   3 |  [1,2,3]
   4 |  [1,2,3,4]
   5 |  [1,2,3,4,5,6]
   6 |  [1,2,3,4,5,6,8,9]
   7 |  [1,2,3,4,5,6,7,8,9,10,12]
   8 |  [1,2,3,4,5,6,7,8,9,10,12,15,16,18]
   9 |  [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,24,27]
  10 |  [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25,27,30,32,36].
		

Links

• Reinhard Zumkeller, Rows n = 0..36 of triangle, flattened

Crossrefs

Cf. A000041, A000792, A034891.

Programs

Haskell

import Data.List (nub, sort)
a212721 n k = a212721_row n !! (k-1)
a212721_row = nub . sort . (map product) . ps 1 where
   ps x 0 = [[]]
   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
a212721_tabf = map a212721_row [0..]

Mathematica

row[n_] := Union[Times @@@ IntegerPartitions[n]];
Table[row[n], {n, 0, 10}] (* Jean-François Alcover, Jun 29 2019 *)

Sage

[sorted(list(set([mul(p) for p in Partitions(n)]))) for n in range(11)] # Peter Luschny, Dec 13 2015