There are multiple ways to parenthesize a sequence of factors two at a time. For instance, , , and can be grouped as either or ; the Catalan numbers count the number of ways this is possible for factors. The groupings form a partially ordered set, where one grouping covers another if the first can be transformed into the second by taking a subelement of the form and replacing it with . In these transformations, , , and can be a single factor or a product of factors. These partially ordered sets form a lattice called the Tamari lattice.[more]
This Demonstration shows graphs formed by lattices generated from factor sequences of different lengths, where each vertex label is a polygon triangulation corresponding to a particular grouping of factors.[less]
R. P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge: Cambridge University Press, 1999 pp. 234–235.