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?

It seems necessary to clarify, that the analogy between $\mathbb{Z}, \mathbb{F}_q[t],\mathbb{C}(z)$ has gotten attention recently and I have not read them closely enough:

- how to count with topology
- Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields
- Statistics of Number Fields and Function Fields

I have found many individual parts of these papers difficult to grasp - and they are put together - and I maybe I can ask more questions on these topics later?

Today, my question may have to do with the last link... suppose we do have this machine comparing statistics on the function field $\mathbb{F}_q[t]$ to statistics of $S_n$ as Qiaochu say. How do we "dequantize" to get a result in $\mathbb{Z}$? The implication is there is some kind of permutation group action on the integers and I was wondering what it could be.

Maybe it's $q \to 1$ limit?