The associahedra are a family of polytopes discovered independently by Tamari and Stasheff. Many models are known; some geometric, some combinatorial.
The vertices correspond to all complete bracketings of an n-fold product, for some given n. The higher-dimensional faces can then be seen as partial bracketings.
A particular attractive model for the vertices is the left bracketing functions (LBFs) of Huang and Tamari. In his thesis, Christopher Nguyen describes the more general higher left bracketing functions (HLBFs), which can be used to describe all the faces.
In this talk I gave an alternative model, in which a face is specified by a pair of LBFs, subject to certain conditions.