  We study the set $S_{ann-nc}$ of permutations of $\{1, ..., p+q \}$ which are
non-crossing in an annulus with $p$ points marked on its external circle and
$q$ points marked on its internal circle. The algebraic approach to
$S_{ann-nc}$ goes by identifying three possible crossing patterns in an
annulus, and by defining a permutation to be annular non-crossing when it does
not display any of these patterns. We prove the annular counterpart for a
``geodesic condition'' shown by Biane to characterize non-crossing permutations
in a disc. We point out that, as a consequence, annular non-crossing
permutations appear in the description of the second order asymptotics for the
joint moments of certain families (Wishart and GUE) of random matrices. We
examine the relation between $S_{ann-nc}$ and the set $NC_{ann}$ of annular
non-crossing partitions of $\{1, ..., p+q \}$, and observe that (unlike in the
disc case) the natural map from $S_{ann-nc}$ onto $NC_{ann}$ has a pathology
which prevents it from being injective.
