General Method to arrange $n^n$, $(n-1)^{(n-1)}$, $(n+1)^{(n+1)}$, $(n-1)^{(n+1)}$, $(n+1)^{(n-1)}$ in ascending order.

Question

Is there a general method to arrange $n^n$, $(n-1)^{(n-1)}$, $(n+1)^{(n+1)}$, $(n-1)^{(n+1)}$, $(n+1)^{(n-1)}$ in ascending order?

I am interested in a general method which works for all real values of n.

Highly original and intuitive methods which may work for specific cases of n also appreciated.

I would also like to understand some beautiful and thought provoking methods(such as proof without words)

Eventually,

My last question is whether we can arrange

$n^n$, $(n-k)^{(n-k)}$, $(n+k)^{(n+k)}$, $(n-k)^{(n+k)}$, $(n+k)^{(n-k)}$ in ascending order.

Where $k$ is a arbitrary positive constant.

Caution : Consider only those values for the above expressions are defined and real.

Answer 1

(Note that the question moved on while I was thinking and typing... I leave this here anyway for interest)

Provided all the quantities are greater than $e$, you have that for $a>b, a^be$, it is clear that $a^bb$ if both are in this range.

The remaining uncertainty is ordering $n^n$ and $(n-k)^{n+k}$ (and I read $k$ as a small offset on $n$, rather than an unconstrained value). $k$ has to be a significant proportion of $n$, in general, to undermine $(n-k)^{n+k}>n^n$. For example (in the integers) for $n=23$ you only need $k<17$ to keep $(n-k)^{n+k}>n^n$ .

So least for $n\ge5$, the initial asked ordering is $$(n-1)^{(n-1)}<(n+1)^{(n-1)}

---

($5$ is the minimum value in the integers for this ordering, but the actual crossover in the reals is extremely close to $n=\pi+1$, above which point $n^n<(n-1)^{(n+1)}$ is true. This threshold value is from (spreadsheet) calculation, not any theoretical assessment).