In literature, I have seen the weighted Hurwitz number $N_{g,n}(d_1 , d_2 \ldots , d_n)$ which are symmetric number and they can be written as double Schur function expansion.

\begin{align} \label{eq:partitionfunction}
Z(\mathbf{p}; \mathbf{q}; h) &= \sum_{\lambda \in {\mathcal P}} s_\lambda(p_1, p_2, \ldots) \, s_\lambda(\tfrac{q_1}{h}, \tfrac{q_2}{h}, \ldots) \, \prod_{\Box \in \lambda} G(c(\Box) h) \notag \\
&= \exp \bigg[ \sum_{g=0}^\infty \sum_{n=1}^\infty \sum_{d_1, d_2, \ldots, d_n = 1}^\infty N_{g,n}(d_1, d_2, \ldots, d_n) \, \frac{h^{2g-2+n}}{n!} \, p_{d_1} p_{d_2} \cdots p_{d_n} \bigg]
\end{align}

where $\bf{p} , \bf{q}$ are parameters,$\mathcal{P}$ set of parition, $G(z)$ formal series, for example $G(z)= \exp(z)$ gives double hurwitz numbers, if we vary $G(z)$ we get different geometrical numbers.

$N_{g,n}(d_1 , \ldots, d_n)$ symmetric polynomials in $\bf{q}$

Question: What condition to be imposed on the family of the symmetric polynomial $C_{g,n}(d_1, \ldots d_n)$ such that the partition function will have double Schur function expansion?

Question: I add an additional constraint that $\sum_{d_1 , \ldots d_n}C_{g,n}(d_1, \ldots d_n)z_1^{d_1}\ldots z_n^{d_n}$ is symmetric meromorphic function in $z_i$ with poles only at finitely many points is it garuentee and double schur function expnasion?