Ergodicity of the adic transformation on the Euler graph 3

The Euler graph has vertices labelled (n, k) for n = 0, 1, 2, ... and k = 0, 1, ..., n, with k + 1 edges from (n, k) to (n + 1, k) and n − k + 1 edges from (n, k) to (n + 1, k + 1). The number of paths from (0,0) to (n, k) is the Eulerian number A(n, k), the number of permutations of 1,2,...,n+ 1 with exactly n− k falls and k rises. We prove that the adic (Bratteli-Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure. 1. The Euler Graph The Euler graph is an infinite directed graph such that at level n there are n + 1 vertices labelled (n, 0) through (n, n). The vertex (n, k) has n+ 2 total edges leaving it, with k + 1 edges connecting it to vertex (n+ 1, k) and n− k + 1 edges connecting it to vertex (n+ 1, k + 1). Define X to be the space of infinite edge paths on the Euler graph. X is a compact metric space in a natural way: if two paths x = x0x1x2... and y = y0y1y2... agree for all n less than j and xj 6= yj , then define d(x, y) = 2 −j . The number of paths from the root vertex (0,0) to the vertex (n, k) is the Eulerian number, A(n, k), which is the number of 2 S. Bailey, M. Keane, K. Petersen, and I. Salama