  Suppose \mu and \mu' are two partitions. We will let \mu \oplus \mu' denote
the "direct sum" of the partitions, defined as the sorted partition made of the
parts of $\mu$ and $\mu'$. In this paper, we define a summation operation on
two Littlewood-Richardson fillings of type (\mu, \nu;\lambda) and (\mu',
\nu';\lambda'), which results in a Littlewood-Richardson filling of type
(\mu\oplus \mu', \nu\oplus \nu' ;\lambda\oplus \lambda'). We give an algorithm
to produce the sum, and show that it terminates in a Littlewood-Richardson
filling by defining a bijection between a Littlewood-Richardson filling and a
flow on a honeycomb, and then showing that the overlay of the two honeycombs of
appropriate type corresponds to the sum of the two fillings.
