The euler adic dynamical system and path counts in the euler graph

Abstract

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 theBratteli-Vershik dynamical system based on the Euler graph is the unique fully supported invariant ergodic Borelprobability 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. © 2010 International Academic Printing Co. Ltd. All rights reserved.