Computational experiments for $1 \le m \le n \le 20$ yield feasible solutions even if you impose the upper bound $a_j \le n$.  Here is an infinite family for $m \le \lceil n/2\rceil +1$:
$$(1,\underbrace{0,\dots,0}_{n-m},2,3,4,\dots,m)$$

Here are the lexicographically smallest solutions for $n=m$:
\begin{matrix}
n & a\\
\hline
1& (1)\\
2& (1, 2)\\
3& (1, 2, 3)\\
4& (1, 3, 4, 2)\\
5& (1, 3, 4, 5, 2)\\ 
6& (1, 3, 6, 5, 4, 2)\\ 
7& (1, 3, 6, 5, 7, 4, 2)\\ 
8& (1, 3, 6, 8, 7, 5, 4, 2)\\ 
9& (1, 3, 6, 8, 7, 9, 5, 4, 2)\\ 
10& (1, 3, 6, 10, 9, 8, 5, 7, 4, 2)\\ 
11& (1, 3, 6, 8, 11, 9, 10, 7, 5, 4, 2)\\ 
12& (1, 3, 6, 11, 9, 12, 10, 7, 5, 8, 4, 2)\\ 
13& (1, 3, 6, 8, 13, 12, 10, 11, 7, 9, 5, 4, 2)\\ 
14& (1, 3, 6, 10, 12, 14, 5, 11, 13, 9, 8, 7, 4, 2)\\ 
15& (1, 3, 6, 8, 14, 12, 15, 11, 13, 9, 7, 10, 5, 4, 2)\\ 
16& (1, 3, 6, 10, 12, 16, 15, 5, 13, 14, 9, 11, 8, 7, 4, 2)\\ 
17& (1, 3, 6, 8, 13, 15, 17, 14, 12, 16, 7, 11, 10, 9, 5, 4, 2)\\ 
18& (1, 3, 6, 8, 13, 16, 18, 17, 15, 14, 7, 10, 12, 11, 9, 5, 4, 2)\\ 
19& (1, 3, 6, 8, 11, 17, 19, 16, 18, 14, 15, 10, 9, 13, 12, 7, 5, 4, 2)\\ 
20& (1, 3, 6, 8, 13, 17, 20, 18, 15, 19, 16, 10, 7, 11, 14, 12, 9, 5, 4, 2)\\ 
\end{matrix}