  We interpret a recent formula for counting orbits of $GL(d,F_q)$ in terms of
counting fixed points as addition in the affine braided line. The theory of
such braided groups (or Hopf algebras in braided categories) allows us to
obtain the inverse relationship, which turns out to be the same formula but
with $q$ and $q^{-1}$ interchanged (a perfect duality between counting orbits
and counting fixed points). In particular, the probability that an element of
$GL(d,F_q)$ has no fixed points is found to be the order-$d$ truncated
$q$-exponential of $-1/(q-1)$.
