AND: multiply OR: add Permutations: order matters Combinations: order does NOT matter Quotient Rule/Identifications/Multisets Inclusion/Exclusion Recursion (generating functions/exponential generating functions)

Defn: The Catalan sequence C_{n} = \frac{1}{n+1} ( \begin{matrix}2n \\ n \end{matrix} )

Thm 8.1.1 Let S_{n} = {a_{1}, a_{2}, ..., a_{2n} | a_{1}, a_{2}, ..., a_{2n} is a sequence containing n +1’s and n -1’s such that Σ_{i=0}^{k}a_{k} ≥ 0 ∀k = 1, 2, ..., 2n}. Then |S_{n}| = C_{n} Ex:

|S_{1}| = |\{1, −1\}| = 1 = C_{1} = \frac{1}{2} ( \begin{matrix}2 \\ 1 \end{matrix} )

|S_{2}| = |\{1, 1, −1, −1; 1, −1, 1, −1\}| = 2 = C_{2} = \frac{1}{3} ( \begin{matrix}4 \\ 2 \end{matrix} )

|S_{3}| = |{1, 1, 1, −1, −1, −1; 1, 1, −1, 1, −1, −1; 1, 1, −1, −1, 1, −1;

1, −1, 1, 1, −1, −1; 1, −1, 1, −1, 1, −1\}| = 5 = C_{3} = \frac{1}{4} ( \begin{matrix}6 \\ 3 \end{matrix} )