Let $S_n$ be the set of $n$th dimentional vectors obtained by permuting $\{1,\ldots,n\}$, $n > 1$.
Let $f_{\textbf{a}}(\textbf{x})=\langle\textbf{a}, \textbf{x}\rangle$, where $\textbf{a}\in\mathbb{N}^n$, be an injection from $S_n$ onto $\mathbb{Z}^+$, and let $p(\textbf{a})=\max f_{\textbf{a}}(\textbf{x})$.
What is $\min p(\textbf{a})$ (in terms of $n$)?
Since $f_{\textbf{a}}(\textbf{x})\in\mathbb{Z}^+$, $\min p(\textbf{a})\ge |S_n|=n!$.
If we have $a_1=0$, and $a_i=n^{i-2}$ for $i>1$, then $f$ is an injection (we would be treating $\textbf{x}$ like digits of a base-$n$ number, with the exception of one number, $x_1$, since given $n-1$ components of $\textbf{x}$ we can always retrieve the last component). This corresponds to $p(\textbf{a})=\sum_{i=2}^{n} i n^{i - 2}$.
So $n! \le \min p(\textbf{a}) \le \sum_{i=2}^{n} i n^{i - 2}$.
This upper bound is not the value of $\min p(\textbf{a})$ in general (using $8^6-1$ instead of $8^6$ for $a_8$ in the $n=8$ case results in an injective map as well), so what is $\min p(\textbf{a})$?