Question

(a) Which permutation of [1,2,3,4,5] follows \(3-1-5-2-4\) in the lexicographical ordering of the permutations of five elements? (b) Repeat the question for \(4-6-1-3-7-5-2\) as a permutation of \([1,2, \ldots, 7\\}\).

Short answer

(a) 3-1-5-4-2 (b) 4-6-1-5-2-3-7

Step-by-step solution

Understanding Lexicographical Order

Lexicographical order compares permutations as if they were words in a dictionary. For numbers, it means examining each element from left to right and comparing them.

Find the Next Permutation for Part (a)

The given permutation is \( 3-1-5-2-4 \). We need to find the next larger permutation. Start from the rightmost position and find the first pair where the first number is less than the next number. Here, it is \(2-4\) (since 2 < 4). Now, find the smallest number greater than 2 to the right of it, which is 4. Swap these two numbers to get \(3-1-5-4-2\). Lastly, reverse the sequence after the swapped position to get \(3-1-5-4-2\).

Result for Part (a)

The permutation \( 3-1-5-4-2 \) is the next in lexicographical order after \( 3-1-5-2-4 \).

Find the Next Permutation for Part (b)

The given permutation is \( 4-6-1-3-7-5-2 \). Again, start from the rightmost position and find the first pair where the first number is less than the next number. Here, it is \(3-7\). Find the smallest number greater than 3 to the right of it, which is 5. Swap these two numbers to get \(4-6-1-5-7-3-2\). Reverse the sequence after the swapped position to get \(4-6-1-5-2-3-7\).

Result for Part (b)

The permutation \( 4-6-1-5-2-3-7 \) is the next in lexicographical order after \( 4-6-1-3-7-5-2 \).

Key concepts

Permutation

A permutation is a specific arrangement of a set of elements. Think of each arrangement as a complete set that has been uniquely ordered. For example, if you have a set consisting of the numbers [1, 2, 3], some possible permutations include [1, 3, 2] and [2, 1, 3].

Permutations are crucial in studies where the order of items matters. For instance:

• Seating arrangements
• Password generation
• Solving puzzles where the sequence is important

When dealing with permutations, it’s important to consider all possible arrangements of the items in the set. For a set with 'n' elements, there are 'n!' (n factorial) permutations. This means a set of 3 elements has 3 × 2 × 1 = 6 permutations. In the exercise above, finding permutations in a specific order is essential for solving the problem step by step.

Next Permutation

Finding the next permutation in lexicographical order is a specific task that ensures you go from the current arrangement to the smallest, next possible arrangement. Imagine you're flipping through the pages of a dictionary and each word needs to be organized by their alphabetical order.

Here's how to find the next permutation:

• Step 1: Identify the longest suffix that is non-increasing. This is the first part of your permutation that needs adjusting.
• Step 2: Find the first number just before this suffix that is smaller than a number within the suffix. This indicates where a change can make the sequence larger.
• Step 3: Swap this number with the smallest number in the suffix that is still larger than it. This makes the sequence closer to a "next" version.
• Step 4: Reverse the numbers in the suffix to get the smallest possible next sequence.

Applying these steps to the sequences given in the original exercise ensures they are perfectly aligned in lexicographical order, which is like arranging them just as one would words in a dictionary.

Discrete Mathematics

Discrete mathematics deals with structures that are distinct and separate. Unlike continuous math that deals with real numbers, discrete mathematics focuses on elements that have specific and distinct values.

This branch of mathematics finds its applications in various fields, including:

• Computer science (algorithms, programming)
• Cryptography (securing data)
• Graph theory (networks, social networks)

Permutations, such as the ones discussed in the exercise, are a core function of discrete mathematics because they deal with finite collections of elements. Understanding discrete math is crucial in breaking down problems with a step-by-step logical approach. This not only assists in computing tasks and algorithmic sequences but also in understanding how systems and different arrangements are structured in various disciplines.