  We give a formula for generalized Eulerian numbers, prove monotonicity of
sequences of certain ratios of the Eulerian numbers, and apply these results to
obtain a new proof that the natural symmetric measure for the Bratteli-Vershik
dynamical system based on the Euler graph is the unique fully supported
invariant ergodic Borel probability measure. Key ingredients of the proof are a
two-dimensional induction argument and a one-to-one correspondence between most
paths from two vertices at the same level to another vertex.
