A statistical sum associated with Young diagrams

Asymptotic properties of the sums of powers of normalized dimensions of all irreducible complex representations of the symmetric group $S_N$ as $N\to\infty$ are studied. The limiting Gibbs measure on the space of Young diagrams is calculated. Our approach is related to a special one-dimensional many particles model. In the appendix written by A.M. Pass some numerical and graphic information on the “Helmholtz energy” for the model is presentede Bibl. – 10.