  We consider arbitrary orderings of the edges entering each vertex of the
(downward directed) Pascal graph. Each ordering determines an adic
(Bratteli-Vershik) system, with a transformation that is defined on most of the
space of infinite paths that begin at the root. We prove that for every
ordering the coding of orbits according to the partition of the path space
determined by the first three edges is essentially faithful, meaning that it is
one-to-one on a set of paths that has full measure for every fully supported
invariant probability measure. We also show that for every $k$ the subshift
that arises from coding orbits according to the first $k$ edges is
topologically weakly mixing. We give a necessary and sufficient condition for
any adic system to be topologically conjugate to an odometer and use this
condition to determine the probability that a random order on a fixed diagram,
or a diagram constructed at random in some way, is topologically conjugate to
an odometer. We also show that the closure of the union over all orderings of
the subshifts arising from codings of the Pascal adic by the first edge has
superpolynomial complexity, is not topologically transitive, and has no
periodic points besides the two fixed points, while the intersection over all
orderings consists of just four orbits.
