Using this$a-b|P(a)-P(b)$ and induction with base case $n=4$ gives that if $\{P(1),P(2),\cdots ,P(n)\}=\{1,2,\cdots, n\}$,then $P(i)=i$ for all $1\le i \le n$ or $P(i)=n+1-i$ for all $1\le i \le n$. So we get $P(x)=R(x)(x-1)\cdots (x-2021)+x$ or $P(x)=S(x)(x-1)\cdots (x-2021)-x+2021$,where $R(x),S(x)\in \mathbb{Z}[x]$