  We resolve an open conjecture from algebraic geometry, which states that two
generating functions for plane partition-like objects (the "box-counting"
formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and
Pandharipande-Thomas theory) are equal up to a factor of MacMahon's generating
function for plane partitions. The main tools in our proof are a
Desnanot-Jacobi-type "condensation" identity, and a novel application of the
tripartite double-dimer model of Kenyon-Wilson.
