A069123 Triangle formed as follows: For n-th row, n >= 0, record the A000041(n) partitions of n; for each partition, write down number of ways to arrange the parts.

1, 1, 2, 1, 6, 2, 1, 24, 6, 4, 2, 1, 120, 24, 12, 6, 4, 2, 1, 720, 120, 48, 24, 36, 12, 6, 8, 4, 2, 1, 5040, 720, 240, 120, 144, 48, 24, 36, 24, 12, 6, 8, 4, 2, 1, 40320, 5040, 1440, 720, 720, 240, 120, 576, 144, 96, 48, 24, 72, 36, 24, 12, 6, 16, 8, 4, 2, 1

Offset: 0

Author

Franklin T. Adams-Watters, Apr 07 2002

Keywords

• easy
• nonn
• tabf

Examples

			This is a function of the individual partitions of an integer. For n = 0 to 5 the terms are (1), (1), (2,1), (6,2,1), (24,6,4,2,1). The partitions are ordered with the largest part sizes first, so the row 4 indices are [4], [3,1], [2,2], [2,1,1] and [1,1,1,1].
.
The irregular table starts:
[0] [1]
[1] [1]
[2] [2, 1]
[3] [6, 2, 1]
[4] [24, 6, 4, 2, 1]
[5] [120, 24, 12, 6, 4, 2, 1]
[6] [720, 120, 48, 24, 36, 12, 6, 8, 4, 2, 1]
		

Links

• 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].

Crossrefs

Cf. A000142, A333144.

Using Abramowitz-Stegun ordering of partitions this becomes array A134133.

Programs

Mathematica

Table[Map[Function[n, Apply[Times, n! ]], IntegerPartitions[i]], {i,0,8}] // Flatten (* Geoffrey Critzer, May 19 2009 *)

SageMath

def A069123row(n):
    return [product(factorial(part) for part in partition) for partition in Partitions(n)]
for n in (0..6): print(A069123row(n)) # Peter Luschny, Apr 10 2020

Formula

[]!=prod_k(n[k]!), or equivalently, []!=prod_k(n[k]!^m[k]).