New properties of almost nilpotent varieties with integer exponents

Almost nilpotent varieties of nonassociative algebras over a field of zero characteristic in the class of all algebras satisfying identical relation $x(yz) \equiv 0$ are studied. Earlier in this class of algebras for each natural number $m \ge 2$ the algebra $A_m$ generating the almost nilpotent variety $var(A_m)$ of exponential growth with exponent of $m$ was defined. In the paper numerical characteristics of varieties $var(A_m)$ are studied.
To this end in the relatively free algebras of the varieties $var(A_m)$ the spaces of multilinear elements corresponding to left normed polynomials with fixed variable on the first position are considered.
Each space is considered as completely reducible module of the symmetric group and multiplicities in the decomposition of the corresponding cocharacter into sum of irreducible characters are calculated. The multiplicities corresponding to the multilinear parts of relatively free algebras of the variety $var(A_m)$ are defined by the calculated values. Colengths of the varieties $var(A_m)$, $m \ge 2$ are obtained using this method. For each $m \ge 2$ the set of identical relations that defines the variety $var(A_m)$ is obtained.