Problem 24 - 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. Four

Step-by-Step Solution

Answer

The number of legally paired sequences with four pairs of left and right parentheses is 14.

Step 1: Identify the value of n

According to the exercise, there are four pairs of left and right parentheses. Therefore, the value of n is 4.

Step 2: Write down the formula

To compute the nth Catalan number, we'll use the formula: $$\mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}$$

Step 3: Plug in the value of n

Now, we'll plug in the value of n=4 into the formula: $$\mathrm{C}_{4}=\frac{(2 \cdot 4) !}{4 !(4+1) !}$$

Step 4: Calculate the factorials

Compute the factorials of the numerators and denominators: $$ 8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 40320$$ $$ 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$$ $$ 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$$

Step 5: Calculate the result using the factorials

Insert the calculated factorials into the formula and compute the final value: $$\mathrm{C}_{4}=\frac{40320}{24 \cdot 120}$$ $$\mathrm{C}_{4}=\frac{40320}{2880}$$ $$\mathrm{C}_{4}=14$$ The number of legally paired sequences with four pairs of left and right parentheses is 14.

Key Concepts

Discrete Mathematics

Discrete Mathematics is a branch of mathematics that deals with discrete objects, that is, those that can be counted as separate and distinct entities. Unlike continuous mathematics, where concepts are often related to physical phenomena with no gaps, discrete mathematics focuses on structures that are fundamentally countable, like integers, graphs, and logical statements.

One key area within discrete mathematics is combinatorics, which involves the study of finite or countable discrete structures. It is a useful tool in computer science for tasks such as analyzing algorithms and counting objects that meet certain criteria. Within combinatorics, we find concepts like the Catalan numbers, which arise in various counting problems connected to recursive structures and combinatorial objects. Understanding how to compute these kinds of numbers is fundamental in discrete mathematics and has practical applications in areas like parsing expressions in computer languages and modeling natural phenomena.

Combinatorics

Combinatorics is a field of study in mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has applications in fields as diverse as physics, computer science, and statistics.

One of the fascinating entities within this field are the Catalan numbers, which represent a sequence of natural numbers that occur in various counting problems. They have a well-established pattern and are instrumental in combinatorial problems like counting distinct binary trees, different ways of associating products in multiplication, and, as in the problem mentioned, the number of ways of legally pairing parentheses.

Real-World Applications

The practicality of combinatorics and Catalan numbers specifically extends to computer science for algorithm designs, such as those for sorting and searching, and understanding the complexity of problems. Moreover, the application of these numbers can be seen not only in theoretical problems but also in gaming strategies and the optimization of networks.

Factorial Notation

Factorial Notation is a mathematical concept used to describe the product of an integer and all the non-zero integers below it. It's symbolized by an exclamation mark (!). For instance, when we see the notation '5!', it means 5 factorial, which is calculated as:

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
Factorial notation is extremely important in combinatorics because it is often used to calculate combinations and permutations – the number of ways of arranging and selecting items without and with regard to the order, respectively. When solving the exercise involving the computation of the Catalan number for four pairs of parentheses, we encountered factorial notation within the derived formula:

\[ \text{Catalan number, } C_{n} = \frac{(2n)!}{(n!)(n+1)!} \]
In the step-by-step solution, we apply factorial notation to compute the necessary values that lead us to the solution. Understanding factorial notation is critical for students in grasping combinatorial concepts, as it frequently emerges in equations and formulas that count combinations and permutations or solve complex problems within discrete mathematics.