Problem 23 - Discrete Mathematics with Applications

Question

The nth Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan ( \(1814-1894 ),\) is defined by $$ \mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0 $$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1 .\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. Three

Step-by-Step Solution

Answer

In conclusion, there are 5 legally paired sequences with 3 pairs of left and right parentheses, as calculated using the Catalan number with n = 3: $$ \mathrm{C}_{3}=\frac{(2 \cdot 3) !}{3 !(3+1) !} = \frac{6!}{3!4!} = \frac{720}{6 \cdot 24} = \frac{720}{144} = 5 $$

Step 1: Compute (2n)!

Since n = 3, we first compute 2n, which is equal to 3 * 2 = 6. Then, we calculate 6! (6 factorial) as follows: $$ 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720 $$

Step 2: Compute n!

Next, we calculate 3! (3 factorial) as follows: $$ 3! = 3 \cdot 2 \cdot 1 = 6 $$

Step 3: Compute (n+1)!

Now, we compute 4! (4 factorial), since n + 1 = 3 + 1 = 4: $$ 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24 $$

Step 4: Apply the formula for the nth Catalan number

We have all of the required values to plug into the formula and find the number of legally paired sequences. We compute the 3rd Catalan number as follows: $$ \mathrm{C}_{3}=\frac{(2 \cdot 3) !}{3 !(3+1) !} = \frac{6!}{3!4!} = \frac{720}{6 \cdot 24} = \frac{720}{144} = 5 $$ In conclusion, there are 5 legally paired sequences with 3 pairs of left and right parentheses.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially useful in the field of combinatorics and the calculation of Catalan numbers. The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). This means you multiply \( n \) by \( n-1 \), \( n-2 \), and so on, all the way down to 1. It's important to remember that by definition, \( 0! = 1 \).

When working with factorials, consider:

• The rapid growth rate of factorials as \( n \) increases. For example, \( 5! = 120 \) and \( 10! = 3,628,800 \).
• Factorials are essential in permutations and combinations, forming the basis of more complex mathematical concepts.
• Factorials allow us to compute the Catalan numbers, which rely on expressions involving products of sequential numbers.

Mastering factorials lays a strong foundation for exploring further into sequences and combinatorial structures like Catalan numbers.

Combinatorics

Combinatorics is the branch of mathematics concerned with counting, arrangement, and combination of objects. It's a field that helps us solve problems related to the choices and arrangements in various scenarios, with applications in computer science, statistics, and beyond.

Within the context of Catalan numbers, combinatorics is key.

• Combinatorics provides the tools needed to understand how different sequences or arrangements can be formed under specific rules.
• The formula for Catalan numbers is a result of combinatorial principles, where factorials play a crucial role.
• Understanding Catalan numbers helps in problems such as arranging parentheses properly, which is vital in syntax parsing in programming.

Grasping combinatorics enables us to unlock numerous possibilities in mathematical sequences and arrangements, making it a powerful tool for solving real-world problems.

Well-Formed Sequences

A well-formed sequence refers to sequences that meet specified rules or criteria, such as correct pairing or arrangement. In the case of Catalan numbers, these sequences are often seen as balanced combinations of elements like parentheses or binary trees.

Well-formed sequences are crucial for understanding Catalan numbers because:

• Catalan numbers directly count these sequences, representing their number for a given set of constraints.
• They emerge frequently in mathematical problems like the sequence of matching parentheses in algebraic expressions.
• Understanding these can also apply to constructing paths in a grid, forming correct expressions, and more.

Exploring well-formed sequences provides insights into how structured patterns are formed, echoing throughout discrete mathematics and beyond.