Symmetry of meromorphic differentials produced by involution identity, and relation to integer partitions

We prove that meromorphic differentials ${\omega }_n^{(0)}(z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n)$ which are recursively generated by an involution identity are symmetric in all their arguments $z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n$. The proof involves an intriguing combinatorial identity between integer partitions into given number of parts.