  We prove various optimal estimates for the counts of friable (i.e. smooth)
permutations and friable polynomials over finite fields.
  In particular, for both permutations and polynomials we prove a result with
an error term that, for the first time, 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.
  Additionally, for permutations, we determine the order of magnitude of
$\log(p_{n,m}/\rho(n/m))$ throughout the range $1\le n \le m$ where $p_{n,m}$
is the probability that a permutation on $n$ elements, chosen uniformly at
random, is $m$-friable.
  We also uncover a phase transition in the polynomial setting. We show that
the probability that a polynomial of degree $n$ in $\mathbb{F}_q$ is
$m$-friable shifts its behavior at $m\approx (3/2)\log_q n$. Previously, phase
transitions were only known to occur at $m\approx 2\log_qn$ and $m\approx
\log_q n$.
