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}