How to enumerate randomly all elements of the lattice ${\Bbb Z}_+^2$ and why Plancherel measure gives the best way for that

V.I. Arnold liked very much (especially during the last period of his activity) combinatorial and asymptotic problems which came from number theory. representation theory, dynamics, and combinatorial topology. According to H.Weyl the fundament of combinatorics based on the deep properties of Symmetric Groups $S_n$ (or analysis of permutations ) and properties of its representations — (e.g. properties of Young diagrams).
Nowadays we know many facts about asymptotic properties of symmetric groups and even about structure of infinite symmetric group. Remarkably this knowledge gives a new look on the finite case. I will illustrate this with new facts about so called Plancherel measure on the Young tableaux and will try to explain why those facts are useful for the formulation of some hypothesis about many dimensional Young tableaux.