Ergodicity of the adic transformation on the Euler graph

The Euler graph has vertices labelled (n, k) for n = 0, 1, 2.... and k = 0, t.... n, with k + 1 edges from (n, k) to (n + 1, k) and n - k + 1, edges frorn (n, k) to (n + 1, k + 1). The number of'paths froin (0,0) to (n, k) is the Euleriall Dumber A(n, k), the number of perinutations oft, 2,..., n+ I with exactly n-k falls and k rises. lVe prove that the adic (Bratteli-ITershik) transformation on the space of infinite paths in this graph is ergodic with respect to the syrnmetric measure.