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So I recently re-encountered the following limit: $$\lim_{n\rightarrow \infty} \dfrac{(n!)^{1/n}}{n}=\dfrac{1}{e}$$

I began to wonder about a few things from this relation:

(i) I notice that $(n!)^{1/n}$ is a sort of "geometric" mean for the factorial; similarly, by dividing by $n$, we are formulating an arithmetic mean. Thus, is there some relation between the standard arithmetic mean (i.e. $x/n$), the standard geometric mean ($x^{1/n}$), the factorial for $n$ and the exponential $e$ ?

(ii) Also, $n!$ can be used to count permutations (granted for finite sets as $n \rightarrow \infty$ so would $n!$). Thus, in some way does this relation reveal some interesting limiting structure of the permutations of the set?

That is, suppose we were to call the real number $(n!)^{1/n}$ for some fixed $n$ to be the "Permutation Root". Now as $n \rightarrow \infty $, this Permutation Root is reaching a limiting value that is $\frac{n}{e}$. In particular, we could also consider the value $a_{n}$ where $$\dfrac{(n!)^{1/n}}{n}=\dfrac{1}{a_{n}} $$ Is there any significance to either of these values (Permuation root or $a_n$)? (Perhaps they are related to the structure of permutations?)

(iii) Now if we were to consider other fields, perhaps even finite-fields/extension fields, would the answers to either of the above questions somehow relate to these algebraic structure? More clearly, can we generalize this exponential-factorial relation, or perhaps some form of this Primitive root/$a_{n}$ notion to some sort of algebraic structure (of groups, fields or whatnot)?

Thoughts: I've become very interested in try to understand the more general structure of the exponential place in the real/complex field; I'm trying to understand if so many of the beautiful properties the relate to this number can some how be further generalized beyond just the real/complex field? I think the above questions sort of inspire this idea. Then again, perhaps (and probably) not. (Oh, and I do recognize that $e$ is a real number; I'm just wondering if we could create a new field with some sort of $e^{*}$ that has similar properties as in the real/complex case, or at least in the context of this formula.)

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I do recognize that $a_{n}$ is the sequence which converges to $e^{-1}$. I'm more interested to understand if this value has some significance to the factorial other than just being part of the sequence which converge to the above limit. –  A.G. Apr 24 '12 at 20:14
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Dear @A.G.: Regarding a combinatorial interpretation - the number $n!/n^n$ is the probability that a randomly chosen endofunction of $\{1, \dots, n\}$ is a permutation. Regards, –  Bruno Joyal Apr 24 '12 at 20:19
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It seems better to think of this value as $$\left(\frac{1}{n}\frac{2}{n}...\frac{n}{n}\right)^{\frac 1 n}$$ –  Thomas Andrews Apr 24 '12 at 20:23
    
@A.G. $1/e$ appears a lot in derangements –  Pedro Tamaroff Apr 24 '12 at 20:24
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Thinking out loud $\ne$ asking a question. –  Gerry Myerson Apr 25 '12 at 5:46
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closed as not a real question by Gerry Myerson, Did, t.b., J. M., Leonid Kovalev Aug 15 '12 at 1:47

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