Question

Define an inversion of a permutation. Define the sign of a permutation. Find the inversion of \(<3,1,4,2>\) Find the sign of \(<3,1,4,2>\).

Short answer

An inversion in a permutation is a pair of elements from the permutation which occur in the opposite of their natural order. Therefore, in the permutation \(<3,1,4,2>\), the inversions are (3,1), (3,2), and (4,2), totaling to 3 inversions. The sign of a permutation is given by \((-1)^n\), where n is the number of inversions in the permutation. In this case, the sign of the permutation \(<3,1,4,2>\) is \((-1)^3 = -1\), indicating a negative permutation. Thus, the inversion of the permutation \(<3,1,4,2>\) is 3 and its sign is negative.

Step-by-step solution

Definition of an Inversion of a Permutation

An inversion of a permutation is a pair of elements from the permutation such that their order in the permutation is the opposite of their natural order. In other words, an inversion occurs when a larger number precedes a smaller number. For example, in the permutation (3,1,4,2), the inversions are (3,1), (3,2), and (4,2).

Calculating the Inversion of a Permutation

To calculate the inversions in the permutation <3,1,4,2>, we will go through each pair of elements and see if they form an inversion. Notice that the inversion pairs are (3,1), (3,2), and (4,2), which are 3 inversions in total.

Definition of the Sign of a Permutation

The sign of a permutation is a value that indicates whether the permutation has an odd or even number of inversions. If the permutation has an odd number of inversions, it has a negative sign, and if it has an even number of inversions, it has a positive sign. Formally, we define the sign of a permutation as follows: sign(P) = (-1)^n, where n is the number of inversions in the permutation P.

Calculating the Sign of a Permutation

Now, we can calculate the sign of the given permutation <3,1,4,2>. We have already found that there are 3 inversions in the permutation. Hence, the sign is (-1)^3 = -1, which means the sign of the permutation <3,1,4,2> is negative. So, the inversion of the permutation <3,1,4,2> is 3, and its sign is negative.

Key concepts

Inversion of a Permutation

An inversion of a permutation is a fundamental concept in combinatorics and provides insight into the nature of permutations. Simply put, whenever you have a sequence of numbers, an inversion occurs when a pair of these numbers is out of their natural order. For example, in a list like (4, 1, 3, 2), if we take the pair (4, 1), the number 4 comes before 1 in the sequence, which is the opposite of their natural numerical order. This is an inversion.

To thoroughly identify inversions, compare each element with the elements that follow it. If any subsequent element is smaller, that pair is marked as an inversion. It's essential for students to understand that while calculating inversions, the position of the elements is crucial. Inversions give us valuable information about the permutation, such as its complexity and how far it is from being sorted.

Sign of a Permutation

The sign of a permutation relates to the concept of inversion by providing an overall statistic about the permutation: whether it has an even or odd number of inversions. This can be a tricky concept to grasp, so let's break it down. The sign can be either +1 or -1, corresponding to even or odd parity, respectively.

To find the sign of a permutation, you would count how many inversions there are in total. If this number is odd, the permutation's sign is negative (represented mathematically as -1). Conversely, if the number is even, the permutation's sign is positive (+1). The mathematical notation for the sign of a permutation P is sign(P) = (-1)^n, where n is the number of inversions. Understanding the sign of a permutation can be particularly useful in advanced areas of mathematics, such as when determining the determinant of a matrix or exploring symmetries in algebraic structures.

Calculating Inversions

Calculating inversions might seem daunting at first, but with a systematic approach, it can be quite simple. Let's consider the permutation <3,1,4,2>. We compare each element with the elements following it in the sequence. When a larger number precedes a smaller one, that pair is counted as an inversion. As we proceed with the example:

• We compare 3 with 1, 4, and 2 and identify two inversions: (3,1) and (3,2).
• Next, compare 1 with 4 and 2—no inversions here, as 1 is smaller than both.
• Following that, compare 4 with 2, which gives us the inversion (4,2).

We found a total of three inversions for the permutation <3,1,4,2>. Such exercises in calculating inversions strengthen a student's combinatorial intuition and prepare them for more complex algebraic concepts.

Parity of Permutations

Parity is a term used to describe whether something is even or odd. In the context of permutations, parity refers to the nature of the permutation based on its number of inversions. This concept is closely linked to the sign of a permutation, as it uses the count of inversions to determine whether the permutation has an even or odd parity.

A permutation with an even number of inversions has even parity and a positive sign, whereas a permutation with an odd number of inversions has odd parity and a negative sign. Understanding the parity of permutations is crucial when studying permutations' properties, such as in group theory and throughout various fields of mathematics like geometry and number theory. Moreover, algorithms for sorting, such as the bubble sort, rely on the concept of parity when counting the number of swaps required to sort a sequence.