A096162 Let n be a number partitioned as n = b_1 + 2*b_2 + ... + n*b_n; then T(n) = (b_1)! * (b_2)! * ... (b_n)!. Irregular triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= A000041(n).

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

Offset: 1

Author

Alford Arnold, Jun 20 2004

Keywords

• easy
• nonn
• tabf

Comments

The partitions of number n are grouped by increasing length and in reverse lexical order for partitions of the same length.

This sequence is in the Abramowitz-Stegun ordering, see A036036. - Hartmut F. W. Hoft, Apr 25 2015

Examples

			Illustrating the formula:
1 1 2 1 3 6 1 4 6 12 24 ... A036038
1 1 1 1 3 1 1 4 3  6  1 ... A036040
so
1 1 2 1 1 6 1 1 2  2 24 ... this sequence.
.
From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)
The sequence as a structured triangle. The column headings indicate the number of elements in the underlying partitions. Brackets indicate groups of the products of factorials for all partitions of the same length when there is more than one partition.
     1   2        3        4     5    6
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
The partitions, their multiplicities and factorial products associated with the five entries in row n = 4 are:
partitions:         {4}, [{3, 1}, {2, 2}], {2, 1, 1}, {1, 1, 1, 1}
multiplicities:      1,  [{1, 1},  2],     {1, 2},     4
factorial products:  1!, [1!*1!, 2!],      1!*2!,      4!
(End)
		

References

• Abramowitz and Stegun, Handbook of Mathematical Functions, p. 831, column "M_1" divided by "M_3."

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

Row sums in A096161.

Row lengths in A000041.

Cf. A036038, A036040, A130561.

Programs

Mathematica

(* function a096162[ ] computes complete rows of the triangle *)
row[n_] := Map[Apply[Times, Map[Factorial, Last[Transpose[Tally[#]]]]]&, GatherBy[IntegerPartitions[n], Length], {2}]
triangle[n_] := Map[row, Range[n]]
a096162[n_] := Flatten[triangle[n]]
Take[a096162[9],90] (* data *)  (*Hartmut F. W. Hoft, Apr 25 2015 *)

SageMath

from collections import Counter
def A096162_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [h(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (1..9): print(A096162_row(n)) # Peter Luschny, Nov 01 2019

Formula

T(n, k) = A036038(n,k) / A036040(n,k).

Appears to be n! / A130561(n); e.g., 4! / (24,24,12,12,1) = (1,1,2,2,24). - Tom Copeland, Nov 12 2017

Extensions

Edited and extended by Christian G. Bower, Jan 17 2006