Analogy between Integers and Permutations 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:


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*$m = p_1\cdots{p_k} \in \mathbb{Z}$ 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:


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*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?
 A: it's possible to extend the analogy to the factorization of polynomials over finite fields $\mathbb{F}_q$ (see this blog post for details; one needs to take $q \to \infty$ for the statistics to match, and in the post I only verify convergence in the sense of moments and am a bit sloppy). In this setting the permutation is Frobenius. But I don't think there's an analogous statement on the number field side. 
I think results like this should be thought of as central limit-type theorems more than anything else; there are certain kinds of statistics that occur universally in certain general situations which otherwise don't necessarily have much in common. 
A: The essential common feature that insures convergence to a Poisson Dirichlet distribution is explained in the book "Logarithmic Combinatorial Structures" by Arratia Barbour and Tavare. They do a great job, I think.
A: If I remember well, there is also a correspondence between the degrees of factors of polynomials in $\mathbb{F}_q[X]$, the size of cycles of riffle-shuffle permutations (a brand of non-uniform random permutations ), and the size of factors in the Lyndon decomposition of random words on an alphabet with $q$ letters, with the Poisson Dirichlet distribution as an asymptotic again. I think I found this in the following paper : Diaconis, M.J. McGrath, and J. Pitman, Riffle shuffles, cycles, and descents, Combinatorica, 1995, in which a correspondence by Ira Gessel was mentioned.
