How many ways are there to express a number as the product of groups of three of its factors?

Question

Specifically, I am thinking of a cuboid with a given volume ($28\,000$) that has sides of integer length. For example, $20 \cdot 20 \cdot 70 = 28\,000$, but so do $10 \cdot 40 \cdot 70$ and $1 \cdot 1 \cdot 28\,000$. I am interested in finding how many possible integer combinations of side lengths there are that produce this volume.

Its prime factorisation is $2^5 \cdot 5^3 \cdot 7$, so I think that the answer may have something to do with permutations of those.

The order of the three groups does matter because there is a distinction between it being height, width or length.

Answer 1

Suppose that $n=p_{1}^{k_1}\cdot p_{2}^{k_2}...p_{m}^{k_m}$. Then what we must do is decide is how to split up each prime power as the sum of $3$ non-negative integers. The way to do that is stars-and-bars and the formula becomes $\binom{k_{i}+2}{2}$. (This gives us the number of ways of splitting up the factor $p_{i}^{k_{i}}$.) All together, this gives us $\prod_{i=1}^{m}\binom{k_{i}+2}{2}$ ways to write $n$ as a product of exactly $3$ of its factors.

To be more explicit in why this works, think of writing your $3$ bins, $\ell, w, h$. $\ell$ will contain $p_{i}^{x_{1}}$, $w$ will contain $p_{i}^{x_2}$ and $h$ will contain $p_{i}^{x_{3}}$. In order for $\ell w h=n$ you need that $p_{i}^{x_1}p_{i}^{x_2}p_{i}^{x_3}=p_{i}^{k_{i}}$ so it must be that $x_1+x_2+x_3=k_i$ and each $x_{1},x_{2},x_{3}\geq 0$. Then, do this for each prime.

Answer 2

Using the formalism of the following MSE link and the Polya Enumeration Theorem it follows that for $n$ having factorization $$n = \prod_{p|n} p^v$$ applying PET we have almost by inspection that the desired count of factorizations into $k$ factors is given by the term $$H(n, k) = \left[\prod_p X_p^v\right] Z(E_k)\left(\prod_p \frac{1}{1-X_p}\right)$$ where the square bracket denotes coefficient extraction of formal power series and $Z(E_k) = a_1^k$ is the cycle index of the identity group containing the identity permutation. This becomes $$\left[\prod_p X_p^v\right] \left(\prod_p \frac{1}{1-X_p}\right)^k.$$ Doing the coefficient extraction we obtain $$\prod_p {v+k-1\choose k-1}$$ as observed by the first responder.