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 $m \le \lceil n/2\rceil +1$: $$(1,\underbrace{0,\dots,0}_{n-m},2,3,4,\dots,m)$$
RobPratt
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