Complexity functions of some Leibniz–Poisson algebras

Leibniz–Poisson algebras are generalizations of Poisson algebras. Let $\{c_n(\mathbf{V})\}_{n\geq 0}$ and $\{\gamma_n(\mathbf{V})\}_{n\geq 2}$ are respectively sequences of codimensions and proper codimensions of varieties of Leibniz-Poisson algebras $\mathbf{V}$. We study the exponential generating functions $\mathcal{C}(\mathbf{V},z)=\sum_{n=0}^{\infty}c_n(\mathbf{V})z^n/n!$ and $\mathcal{C}^{p}(\mathbf{V},z)=\sum_{n=2}^{\infty}\gamma_n(\mathbf{V})z^n/n!$. The functions $\mathcal{C}(\mathbf{V},z)$ are used in the study of Lie algebras and associative algebras. In this paper we study numerical characteristics of varieties of Leibniz–Poisson algebras $\mathbf{V}_s$ defined by the identities
$$ \{ x_1, x_2 \} \cdot \{x_3, x_4 \} =0, ~\{x_0,\{x_1,x_2\},\ldots ,\{x_{2s-1},x_{2s}\}\}=0 $$
and of varieties of Leibniz–Poisson algebras $\mathbf{W}_s$ defined by the identities
$$ \{ x_1, x_2 \} \cdot \{x_3, x_4 \} =0, ~\{\{x_1,x_2\},\ldots ,\{x_{2s+1},x_{2s+2}\}\}=0, ~s\geq 1. $$
For each of the variety $\mathbf{V}_s$ and $\mathbf{W}_s$ an algebra-carrier is found and a basis of $n$-th proper polylinear space is built. We found exact formulas for the exponential generating functions for the codimension sequences and for the proper codimension sequences and exact formulas for codimension and proper codimension. Also a series of varieties of Leibniz–Poisson algebras, which codimension sequences asymptotically grow as polynomials of degree $k$, $k \geq 2 $, is given.