  In this paper, we present the results of a computer investigation of
asymptotics for maximum dimensions of linear and projective representations of
the symmetric group. This problem reduces to the investigation of standard and
strict Young diagrams of maximum dimensions. We constructed some sequences for
both standard and strict Young diagrams with extremely large dimensions. The
conjecture that the limit of normalized dimensions exists was proposed 30 years
ago [A.~M.~Vershik and S.~V.~Kerov, 1985] and has not been proved yet. We
studied the growth and oscillations of the normalized dimension function in
sequences of Young diagrams. Our approach is based on analyzing finite
differences of their normalized dimensions. This analysis also allows us to
give much more precise estimation of the limit constants.
