A333144 Irregular triangle where row n lists the product of the factorials of the exponentials of the partitions of n and the partitions are enumerated in canonical order.

1, 1, 1, 2, 1, 1, 6, 1, 1, 2, 2, 24, 1, 1, 1, 2, 2, 6, 120, 1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720, 1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040, 1, 1, 1, 2, 1, 1, 6, 2, 1, 2, 2, 24, 2, 4, 2, 6, 120, 24, 12, 48, 720, 40320

Offset: 0

Author

Peter Luschny, Apr 10 2020

Keywords

• nonn
• tabf

Comments

By 'canonical order' we understand the graded reverse lexicographic order (the default order of Mathematica and SageMath).

Examples

			The irregular table starts:
[0] [1]
[1] [1]
[2] [1, 2]
[3] [1, 1, 6]
[4] [1, 1, 2, 2, 24]
[5] [1, 1, 1, 2, 2, 6, 120]
[6] [1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720]
[7] [1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040]
		

Crossrefs

Row sums are A161779.

Cf. A069123.

Programs

SageMath

def A333144row(n):
    return [product(factorial(expo) for expo in partition.to_exp()) for partition in Partitions(n)]
for n in (0..9): print(A333144row(n))