Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics

Vertically symmetric alternating sign matrices (VSASMs) of order $2n+1$ are known to be equinumerous with lozenge tilings of a hexagon with side lengths $2n+2,2n,2n+2,2n,2n+2,2n$ and a central triangular hole of size $2$ that exhibit a cyclical as well as a vertical symmetry, but no bijection between these two classes of objects has been constructed so far. In order to make progress towards finding such a bijection, we generalize this result by introducing certain natural extensions for both objects along with $n+3$ parameters and show that the multivariate generating functions with respect to these parameters coincide. The equinumeracy of VSASMs and the lozenge tilings is then an easy consequence of this result, which is obtained by specializing the generating functions to signed enumerations for both types of objects. In fact, we present several versions of such results (one of which was independently conjectured by Florian Aigner) but in all cases certain natural extensions of the original objects are necessary and that may hint at why it is so hard to come up with an explicit bijection for the original objects.