  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.
