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 of the Temperley–Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ2-weighted punctured cyclically symmetric transpose complement plane partitions where τ=−(q+q−1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ2-enumerations of vertically symmetric alternating sign matrices and modifications thereof.