  Vershik and Kerov conjectured in 1985 that dimensions of irreducible
representations of finite symmetric groups, after appropriate normalization,
converge to a constant with respect to the Plancherel family of measures on the
space of Young diagrams. The statement of the Vershik-Kerov conjecture can be
seen as an analogue of the Shannon-McMillan-Breiman Theorem for the
non-stationary Markov process of the growth of a Young diagram. The limiting
constant is then interpreted as the entropy of the Plancherel measure. The main
result of the paper is the proof of the Vershik-Kerov conjecture. The argument
is based on the methods of Borodin, Okounkov and Olshanski.
