  We study the Kronecker coefficients $g_{\lambda, \mu, \nu}$ via a formula
that was described by Mishna, Rosas, and Sundaram, in which the coefficients
are expressed as a signed sum of vector partition function evaluations. In
particular, we use this formula to determine formulas to evaluate, bound, and
estimate $g_{\lambda, \mu, \nu}$ in terms of the lengths of the partitions
$\lambda, \mu$, and $\nu$. We describe a computational tool to compute
Kronecker coefficients $g_{\lambda, \mu, \nu}$ with $\ell(\mu) \leq 2,\
\ell(\nu) \leq 4,\ \ell(\lambda) \leq 8$. We present a set of new vanishing
conditions for the Kronecker coefficients by relating to the vanishing of the
related atomic Kronecker coefficients, themselves given by a single vector
partition function evaluation. We give a stable face of the Kronecker
polyhedron for any positive integers $m,n$. Finally, we give upper bounds on
both the atomic Kronecker coefficients and Kronecker coefficients.
