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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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closed as not a real question by Gerry Myerson, Did, t.b., J. M., Leonid Kovalev Aug 15 '12 at 1:47

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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
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
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
Thinking out loud $\ne$ asking a question. –  Gerry Myerson Apr 25 '12 at 5:46