We define integral measures of complexity for Heegaard splittings based on the
graph dual to the curve complex and on the pants complex defined by Hatcher and Thurston.
As the Heegaard splitting is stabilized, the sequence of complexities turns out to converge
to a non-trivial limit depending only on the manifold. We then use a similar method to
compare different manifolds, defining a distance which converges under stabilization to an
integer related to Dehn surgeries between the two manifolds.