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# Journal of Combinatorics

## Volume 6 (2015)

### Number 3

### Flows on honeycombs and sums of Littlewood–Richardson tableaux

Pages: 353 – 394

DOI: http://dx.doi.org/10.4310/JOC.2015.v6.n3.a6

#### Authors

#### Abstract

Suppose $\mu$ and $\mu^\prime$ are two partitions. We will let $\mu \oplus \mu^\prime$ denote the *direct sum* of the partitions, defined as the sorted partition made of the parts of $\mu$ and $\mu^\prime$. In this paper, we define a summation operation on two Littlewood–Richardson fillings of type $(\mu , \nu ; \lambda)$ and $(\mu^\prime , \nu^\prime ; \lambda^\prime)$, which results in a Littlewood–Richardson filling of type $(\mu \oplus \mu^\prime , \nu \oplus \nu^\prime ; \lambda \oplus \lambda^\prime)$. 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.