- #1
johnnyboy2005
- 29
- 0
i was wondering if anyone can help me understand why the product of two odd permutations is odd? i came across this on a website but it didn't help me understand why. thanks for the help
Be careful: I don't mean swapping what is in the first and second places: I specifically meant swapping the symbols 1 and 2. The 3 in the first place and the 4 in the last place don't change. 1 moves from 2nd to 3rd place while 2 moves from 3rd to 2nd place.first swap 1 and 3: 1234-> 3214, then swap 1 and 2: 3214->3124.
can u please explain how swapping 1 and 2 changes 3214 into 3124?
In a model where each permutation of n digits is represented by a node and where there is an edge between two nodes A and B iff there is a single transposition of the permutation of A that transforms it to B. Then you can talk about the lengths of permutations as the length of the paths from an initial permutation A to a final permutation B.matt grime said:At the risk of unnecessarily introducing another (unnecessary) idea (I stil can't tell what Job's getting at)...
An even permutation is a mathematical concept that refers to a rearrangement or reordering of a set of elements. It is considered "even" when it can be achieved by an even number of swaps or transpositions.
To determine if a permutation is even or odd, you can use the parity rule. Count the number of inversions (pairs of elements that are out of order) in the permutation. If the number of inversions is even, the permutation is even; if it is odd, the permutation is odd.
Even and odd permutations are important in various areas of mathematics, such as group theory and abstract algebra. They also have practical applications in fields like cryptography and computer science.
Yes, an odd permutation can be transformed into an even permutation by swapping any two elements. This is because swapping two elements creates an even number of inversions, changing the parity of the permutation.
Even and odd permutations are used in various real-world problems, such as in coding theory to ensure error-free data transmission, in cryptography to create secure encryption methods, and in game theory to analyze strategic decision-making.