Problem 24 - Discrete Mathematics with Applications

Question

The \(n\) th 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 for four pairs of left and right parentheses is 14.

Step 1: Substitute the value of n in the given formula

We are asked to find the number of legally paired sequences for four pairs of parentheses. To do this, we will plug in n=4 into the given formula for the nth Catalan number: $$\mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}$$

Step 2: Calculate the factorial values

Now, we'll calculate the factorials needed for the formula: $$2n! = (2 * 4)! = 8!$$ $$n! = 4!$$ $$(n+1)! = (4+1)! = 5!$$

Step 3: Compute the Catalan number

Now, substitute these values into the given formula and compute the Catalan number: $$\mathrm{C}_{4}=\frac{8!}{4!(5!)}$$ To simplify, we can write the factorials as a product of their integer values: $$\mathrm{C}_{4}=\frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2}{(4 \cdot 3 \cdot 2 \cdot 1)(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$$ Now, cancel out common terms from the numerator and denominator: $$\mathrm{C}_{4}=\frac{8 \cdot 7 \cdot 6}{(5 \cdot 4 \cdot 3 \cdot 2)}$$ Finally, compute the value: $$\mathrm{C}_{4}=\frac{8 \cdot 7 \cdot 6}{(5 \cdot 4 \cdot 3 \cdot 2)} = 14$$

Step 4: Interpret the result

The value of the fourth Catalan number, \(\mathrm{C}_{4}\), is 14. This means that there are 14 different legally paired sequences of four pairs of left and right parentheses.

Key Concepts

Discrete Mathematics

Discrete mathematics is an area of mathematics that is concerned with study of discrete and often finite objects. It includes a variety of topics, such as logic, set theory, graph theory, and counting, which are essential for computer science, cryptography, and network analysis.

Within this field, Catalan numbers form an intriguing sequence of natural numbers that have many applications in various discrete mathematics problems, including the enumeration of specific types of lattice paths, structures in combinatorics, and even in sorting algorithms. Understanding the way these numbers are computed and how they can be applied to computational problems is a fundamental aspect of discrete mathematics that bridges the divide between abstract theory and practical application.

Factorials

In mathematics, a factorial is a function that multiplies a number by every positive integer less than it. Notated as 'n!', it is foundational to many areas of mathematics, including permutations and combinations in combinatorics.

The factorial function grows rapidly with increasing values of n, which makes it especially interesting when evaluating expressions like the Catalan number formula. Factorial expressions often involve a high degree of symmetry and cancellation when manipulated algebraically, as in the case of computing Catalan numbers, where multiple factorial terms are involved. Understanding how to simplify factorials within formulas helps in solving many problems in discrete mathematics and related fields.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results. It deals with the study of finite or countable discrete structures and includes problems related to counting the number of certain arrangements or sequences that meet specific criteria.

The Catalan numbers are a classic subject in combinatorics because they enumerate various problems such as the number of ways to correctly pair parentheses, the number of rooted binary trees with a certain number of nodes, and even the possible paths on a grid under certain conditions. In our exercise example, combinatorics is used to determine the number of distinct ways to arrange four pairs of parentheses, which turns out to be the fourth Catalan number. This demonstrates the power of combinatorial thinking: by breaking down complex problems into countable structures, we can employ mathematical tools to find precise solutions.