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Unformatted text preview: Catalan Numbers Tom Davis [email protected] http://www.geometer.org/mathcircles November 24, 2010 We begin with a set of problems that will be shown to be completely equivalent. The solution to each problem is the same sequence of numbers called the Catalan numbers. Later in the document we will derive relationships and explicit formulas for the Catalan numbers in many different ways. 1 Problems 1.1 Balanced Parentheses Suppose you have n pairs of parentheses and you would like to form valid groupings of them, where “valid” means that each open parenthesis has a matching closed parenthesis. For example, “ (()()) ” is valid, but “ ())()( ” is not. How many groupings are there for each value of n ? Perhaps a more precise definition of the problem would be this: A string of parentheses is valid if there are an equal number of open and closed parentheses and if you begin at the left as you move to the right, add 1 each time you pass an open and subtract 1 each time you pass a closed parenthesis, then the sum is always nonnegative. Table 1 shows the possible groupings for ≤ n ≤ 5 . n = 0 : * 1 way n = 1 : () 1 way n = 2 : ()(), (()) 2 ways n = 3 : ()()(), ()(()), (())(), (()()), ((())) 5 ways n = 4 : ()()()(), ()()(()), ()(())(), ()(()()), ()((())), 14 ways (())()(), (())(()), (()())(), ((()))(), (()()()), (()(())), ((())()), ((()())), (((()))) n = 5 : ()()()()(), ()()()(()), ()()(())(), ()()(()()), ()()((())), 42 ways ()(())()(), ()(())(()), ()(()())(), ()((()))(), ()(()()()), ()(()(())), ()((())()), ()((()())), ()(((()))), (())()()(), (())()(()), (())(())(), (())(()()), (())((())), (()())()(), (()())(()), ((()))()(), ((()))(()), (()()())(), (()(()))(), ((())())(), ((()()))(), (((())))(), (()()()()), (()()(())), (()(())()), (()(()())), (()((()))), ((())()()), ((())(())), ((()())()), (((()))()), ((()()())), ((()(()))), (((())())), (((()()))), ((((())))) Table 1: Balanced Parentheses * It is useful and reasonable to define the count for n = 0 to be 1 , since there is exactly one way of arranging zero parentheses: don’t write anything. It will become clear later that this is exactly the right interpretation. 1 1.2 Mountain Ranges How many “mountain ranges” can you form with n upstrokes and n downstrokes that all stay above the original line? If, as in the case above, we consider there to be a single mountain range with zero strokes, Table 2 gives a list of the possibilities for ≤ n ≤ 3 : n = 0 : * 1 way n = 1 : /\ 1 way n = 2 : /\ 2 ways /\/\, / \ n = 3 : /\ 5 ways /\ /\ /\/\ / \ /\/\/\, /\/ \, / \/\, / \, / \ Table 2: Mountain Ranges Note that these must match the parenthesisgroupings above. The “ ( ” corresponds to “ / ” and the “ ) to “ \ ”. The mountain ranges for n = 4 and n = 5 have been omitted to save space, but there are 14 and 42 of them, respectively. It is a good exercise to draw the 14 versions with n = 4 ....
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 Fall '09
 DR.KARENDANIELS

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