Braided line and counting fixed points of GL(d, double struck F sign _q)

We interpret a recent formula for counting orbits of GL(d, double struck F sign_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, double struck F sign_q) has no fixed points is found to be the truncated q-exponential.