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

1,3

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

References

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

Links

Table of n, a(n) for n=1..90.

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].

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

Example

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)

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 *)

Prog

(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

Crossrefs

Row sums in A096161.

Row lengths in A000041.

Cf. A036038, A036040, A130561.

Sequence in context: A320637 A216919 A152656 * A333144 A306297 A053383

Adjacent sequences: A096159 A096160 A096161 * A096163 A096164 A096165

Keyword

easy,nonn,tabf

Author

Alford Arnold, Jun 20 2004

Extensions

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

Status

approved