Smooth permutations and polynomials revisited

We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The main term is the usual Dickman $\rho$ function, but with its argument shifted. We determine the order of magnitude of $\log(p_{n,m}/\rho(n/m))$ where $p_{n,m}$ is the probability that a permutation on $n$ elements, chosen uniformly at random, is $m$-smooth. We uncover a phase transition in the polynomial setting: the probability that a polynomial of degree $n$ in $\mathbb{F}_q$ is $m$-smooth changes its behavior at $m\approx (3/2)\log_q n$.