Need literature or link to family of cryptographic invertible polynomials which has distribution similar to inverted bell curve
These requirements are impossible. If $p_k$ is a permutation, then the distribution induced by "choose $x$ uniformly in $\mathbb{Z}_n$, output $y=p_k(x)$" is also uniform in  $\mathbb{Z}_n$. It can't follow a normal distribution. For every possible $y$, there is exactly one $x$ that makes it happen, and that $x$ is assigned probability $1/2^n$.