I am reading Andrew Granville's Anatomy of Integers and Permutations where it is argued the factorization of a permutation into disjoint cycles is analogous to the factorization of a number into prime factors.

In the blog-sphere you can find these two ways of defining partitions of unity:

• $m = p_1\cdots{p_k} \in \mathbb{Z}$ and $\{ \frac{\log p_1}{\log m}, \dots, \frac{\log p_k}{\log m}\}$.
• $\sigma = C_1\dots C_k \in S_n$ and $\{ \frac{|C_1|}{n}, \dots, \frac{|C_n|}{n} \} $

One can prove both of these converge to the Poisson-Dirichlet process. It looks like $n \approx \log m$ in this analogy and $\mathbb{Z} \simeq S_n $.

This is a correspondence between partitions, but what could be permuted in the $\mathbb{Z}$ side?