One-dimensional central measures on numberings of ordered sets

We describe one-dimensional central measures on numberings (tableaux) of ideals of partially ordered sets (posets). As the main example, we study the poset ${\mathbb{Z}}_{+}^d$ and the graph of its finite ideals, multidimensional Young tableaux; for $d=2$, it is the ordinary Young graph. The central measures are stratified by dimension; in the paper we give a complete description of the one-dimensional stratum and prove that every ergodic central measure is uniquely determined by its frequencies. The suggested method, in particular, gives the first purely combinatorial proof of E.~Thoma’s theorem for one-dimensional central measures different from the Plancherel measure (which is of dimension~$2$).