There is a question in my textbook:
Find a direct one-to-one mapping from the n-gons of Question 3 to the products of Example 14.7.
Question 3 asks: In how many ways can one decompose a convex n-gon into triangles by n − 3 nonintersecting diagonals?
Example 14.7 states: Suppose that we have a set S with a nonassociative product operation. An expression $x_1x_2 ...x_n$ with $x_i ∈ S$ does not make sense, and brackets are needed to indicate the order in which the operations are to be carried out. Let $u_n$ denote the number of ways to do this if there are n factors $x_i$. For example, $u_4 = \frac{1}{4} {6 \choose 3} = 5$ corresponding to the products (a(b(cd))), (a((bc)d)), ((ab)(cd)), ((a(bc))d), and (((ab)c)d). Each product contains within the outer brackets two expressions, the first a product of m factors and the second a product of $n − m$ factors, where $1 ≤ m ≤ n − 1$.
I do not understand what question 3 is asking. I understand example 14.7, so to answer this question I am to make a 1-1 mapping from these n-gons to nonassociative products, but this is impossible and incredibly frustrating as I do not understand question 3. Can someone please clarify?
EDIT: I found this that is helping me understand: https://www.math.ucla.edu/~pak/lectures/Cat/Van_Lint-1-4_25-27.pdf However I don't see how to make a 1-1 mapping between the two.