  Kostant's partition function is a vector partition function that counts the
number of ways one can express a weight of a Lie algebra $\mathfrak{g}$ as a
nonnegative integral linear combination of the positive roots of
$\mathfrak{g}$. Multiplex juggling sequences are generalizations of juggling
sequences that specify an initial and terminal configuration of balls and allow
for multiple balls at any particular discrete height. Magic multiplex juggling
sequences generalize further to include magic balls, which cancel with standard
balls when they meet at the same height. In this paper, we establish a
combinatorial equivalence between positive roots of a Lie algebra and throws
during a juggling sequence. This provides a juggling framework to calculate
Kostant's partition functions, and a partition function framework to compute
the number of juggling sequences. From this equivalence we provide a broad
range of consequences and applications connecting this work to polytopes,
posets, positroids, and weight multiplicities.
