Tokyo Journal of Mathematics

The Euler Adic Dynamical System and Path Counts in the Euler Graph


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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.

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

First available in Project Euclid: 31 January 2011

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Zentralblatt MATH identifier

Primary: 37A05: Measure-preserving transformations 37A25: Ergodicity, mixing, rates of mixing 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx] 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
Secondary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 37A55: Relations with the theory of C-algebras [See mainly 46L55]


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.

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