# Why does the Dyson series have a 1/n! factor?

This is the explanation from Wikipedia:

Is there a more rigorous proof or explanation of how reducing the integration region to these sub-regions introduces a $$\frac{1}{n!}$$ factor? I am confused about why the number of these sub-regions is $$n!$$ in the first place. They way the sub-regions are defined it would seem as if there are $$n$$ of them.

The $$\frac{1}{n!}$$ factor and the fact that $$K$$ has to be symmetric suggests that the proof involves looking at different permutations of the time boundaries $$t_1, t_2,$$ etc. However, the proof given here (and other proofs I've found) never show this step explicitly.

There are $$n$$ time variables. Each way of ordering the times is a permutation of the $$n$$ variables. There are $$n!$$ permutations of $$n$$ time variables ($$n$$ ways to choose the highest time, $$n-1$$ ways to choose the second highest time, etc.).
• @pll04 Have you done it for $n=2$? Commented Feb 27 at 16:31
• @pll04 Since you've done it for $n=2$, repeat for $n=3$. That'll give you a feel for using induction to do $m=n+1$, and then you're set.