I am reading Andrew Granville's [Anatomy of Integers and Permutations](http://www.dms.umontreal.ca/~andrew/PDF/Anatomy.pdf) where it is argued the factorization of a permutation into cycles is analogous with the factorization of a number into prime factors.  

In the [blog-sphere](http://terrytao.wordpress.com/2013/09/21/the-poisson-dirichlet-process-and-large-prime-factors-of-a-random-number/) you can find these two ways of definition partitions of unity:

* $m = p_1, \dots, p_k \in \mathbb{Z}$ 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](http://www.stats.ox.ac.uk/~griff/pd.pdf).  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?