We study almost nilpotent varieties of algebras over a field of zero characteristic. Earlier in the class of algebras with identical relation x(yz) ≡ 0 and in the class of all commutative metabelian algebras countable sets of varieties with integer PI-exponent were defined. Only the existence of almost nilpotent subvariety in each defined variety was proved. In the paper by means of combinatorial methods and methods of the representation theory of symmetric groups we prove that earlier defined varieties are almost nilpotent.
