## Tokyo Journal of Mathematics

### 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 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.

#### Article information

Source
Tokyo J. of Math. Volume 33, Number 2 (2010), 327-340.

Dates
First available in Project Euclid: 31 January 2011

https://projecteuclid.org/euclid.tjm/1296483473

Digital Object Identifier
doi:10.3836/tjm/1296483473

Mathematical Reviews number (MathSciNet)
MR2779260

Zentralblatt MATH identifier
1213.37008

#### Citation

PETERSEN, Karl; VARCHENKO, Alexander. The Euler Adic Dynamical System and Path Counts in the Euler Graph. Tokyo J. of Math. 33 (2010), no. 2, 327--340. doi:10.3836/tjm/1296483473. https://projecteuclid.org/euclid.tjm/1296483473

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