Ordered Factorization
An ordered factorization is a factorization (not necessarily into prime factors) in which 
is considered distinct from 
. The following table lists the ordered factorizations for the integers 1 through 10.
 |
# |
ordered factorizations |
| 1 |
1 |
1 |
| 2 |
1 |
2 |
| 3 |
1 |
3 |
| 4 |
2 |
 , 4 |
| 5 |
1 |
5 |
| 6 |
3 |
 ,  , 6 |
| 7 |
1 |
7 |
| 8 |
4 |
 ,  ,  , 8 |
| 9 |
2 |
 , 9 |
| 10 |
3 |
 ,  , 10 |
The numbers of ordered factorizations 
of 
, 2, ... are given by 1, 1, 1, 2, 1, 3, 1, 4, 2, 3, ... (OEIS A074206). This sequence has the Dirichlet generating function
 |
(1)
|
where 
is the Riemann zeta function.
A recurrence equation for 
is given by
 |
(2)
|
where the sum is over the divisors of 
and 
(Hille 1936, Knopfmacher and Mays 2006). Another recurrence also due to Hille (1936) for 
is given by
![H(n)=2[sum_(p_i)H(n/(p_i))-sum_(p_1,p_2)H(n/(p_ip_j))+...+(-1)^(r-1)H(n/(p_1p_2...p_r))],](https://mathworld.wolfram.com/images/equations/OrderedFactorization/NumberedEquation3.gif) |
(3)
|
where 
and
 |
(4)
|
is the prime factorization of 
(Knopfmacher and Mays 1996).
MacMahon (1893) derived the beautiful double sum formula
 |
(5)
|
where
 |
(6)
|
(Knopfmacher and Mays 1996). In the case that 
is a product of two prime powers,
 |
(7)
|
Chor et al. (2000) showed that
where 
is a hypergeometric function.
The number of ordered factorizations of 
is equal to the number of perfect partitions of 
(Goulden and Jackson 1983, p. 94).
REFERENCES:
Chor, B.; Lemke, P.; and Mador, Z. "On the Number of Ordered Factorizations of Natural Numbers." Disc. Math. 214, 123-133, 2000.
Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 126, 1974.
Goulden, I. P. and Jackson, D. M. Problem 2.5.12 in Combinatorial Enumeration. New York: Wiley, p. 94, 1983.
Hille, E. "A Problem in 'Factorisatio Numerorum.' " Acta Arith. 2, 134-144, 1936.
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 141, 1985.
Knopfmacher, A. and Mays, M. "Ordered and Unordered Factorizations of Integers." Mathematica J. 10, 72-89, 2006.
MacMahon, P. A. "Memoir on the Theory of the Compositions of Numbers." Philos. Trans. Roy. Soc. London (A) 184, 835-901, 1893.
Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 124, 1980.
Sloane, N. J. A. Sequence A074206 in "The On-Line Encyclopedia of Integer Sequences."
Warlimont, R. "Factorisatio Numerorum with Constraints." J. Number Th. 45, 186-199, 1993.
