Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of ${\tau }^2$-weighted punctured cyclically symmetric transpose complement plane partitions where $\tau =-(q+q^{-1})$. In the cases of no or minimal punctures, we prove that these generating functions coincide with ${\tau }^2$-enumerations of vertically symmetric alternating sign matrices and modifications thereof.