  The type $A$ Kostant partition function is an important combinatorial object
with various applications: it counts integer flows on the complete directed
graph, computes Hilbert series of spaces of diagonal harmonics, and can be used
to compute weight and tensor product multiplicities of representations. In this
paper we study asymptotics of the Kostant partition function, improving on
various previously known lower bounds and settling conjectures of O'Neill and
Yip. Our methods build upon recent results and techniques of
Br\"and\'en-Leake-Pak, who used Lorentzian polynomials and Gurvits' capacity
method to bound the number of lattice points of transportation and flow
polytopes. Finally, we also give new two-sided bounds using the Lidskii
formulas from subdivisions of flow polytopes.
