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

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