I will start by discussing a famous open problem in combinatorics: the enumeration of meanders. This problem can be formulated as an optimization problem on the set of non-crossing partitions, that we shall in turn relate to various random matrix models involving tensor products. Finally, these same models are related to a quantum generalization of the max-flow / min-cut duality in graph theory. The goal is this talk (which is meant to be pedagogical) is to introduce and connect these topics, and to present some of the tools (mostly combinatorial) used in these fields.
