  We use Gelfand-Tsetlin diagrams to write down the weight multiplicity
function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition
function. This allows us to apply known results about partition functions to
derive interesting properties of the weight diagrams. We relate this
description to that of the Duistermaat-Heckman measure from symplectic
geometry, which gives a large-scale limit way to look at multiplicity diagrams.
We also provide an explanation for why the weight polynomials in the boundary
regions of the weight diagrams exhibit a number of linear factors. Using
symplectic geometry, we prove that the partition of the permutahedron into
domains of polynomiality of the Duistermaat-Heckman function is the same as
that for the weight multiplicity function, and give an elementary proof of this
for sl_4(C) (A_3).
