arXiv subjects
Publications and source records attributed to K. Petersen.
We obtain formulas for the growth rate of the numbers of certain paths in infinite graphs built on the two-dimensional Eulerian graph. Corollaries are identities relating Stirling numbers of the first and second kinds.
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.