The ${\mathbb{Z}}_2$-graded dimensions of the free Jordan superalgebra $J(D_1\mathrm{\mid }{D}_2)$

Let $k$ be a field of characteristic $0$. For a superspace V=V_\bar{0}\oplus V_\bar{1} over $k$, we call the vector (\dim_k V_\bar{0} ,\dim_k V_\bar{1}) the (${\mathbb{Z}}_2$-)graded dimension of $V$. Let $J(D_1\mathrm{\mid }{D}_2)$ be the free Jordan superalgebra generated by $D_1$ even generators and $D_2$ odd generators. In this paper, we study the graded dimensions of the $n$-components of $J(D_1\mathrm{\mid }{D}_2)$ and find the connection between them and the homology of Tits-Allison-Gao Lie superalgebra of $J(D_1\mathrm{\mid }{D}_2)$ following the method given by I.Kashuba and O.Mathieu in [KM], where they deal with the free Jordan algebra. And, four interesting conjectures of above contents are proposed in our paper.