  Halved monotone triangles are a generalisation of vertically symmetric
alternating sign matrices (VSASMs). We provide a weighted enumeration of halved
monotone triangles with respect to a parameter which generalises the number of
$-1$s in a VSASM. Among other things, this enables us to establish a generating
function for vertically symmetric alternating sign trapezoids. Our results are
mainly presented in terms of constant term expressions. For the proofs, we
exploit Fischer's method of operator formulae as a key tool.
