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Thread: Even vs. Odd Permutations

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    Even vs. Odd Permutations

    I don’t understand how permutations can be even or odd. So take the group S4, which has 24 permutations, 12 odd, and 12 even.

    OK. What is that? How do you determine whether a given permutation is odd or even?

    TIA!

    SLD

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    Learn something new every day. There are videos.

    https://www.bing.com/videos/search?q...4870&FORM=VIRE

    https://en.wikipedia.org/wiki/Parity_of_a_permutation

    This looks like a toy I member as a kid

    https://en.wikipedia.org/wiki/15_puzzle

    The 15-puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others) is a sliding puzzle that consists of a frame of numbered square tiles in random order with one tile missing. The puzzle also exists in other sizes, particularly the smaller 8-puzzle. If the size is 3×3 tiles, the puzzle is called the 8-puzzle or 9-puzzle, and if 4×4 tiles, the puzzle is called the 15-puzzle or 16-puzzle named, respectively, for the number of tiles and the number of spaces. The object of the puzzle is to place the tiles in order by making sliding moves that use the empty space.

    The n-puzzle is a classical problem for modelling algorithms involving heuristics. Commonly used heuristics for this problem include counting the number of misplaced tiles and finding the sum of the taxicab distances between each block and its position in the goal configuration.[1] Note that both are admissible, i.e. they never overestimate the number of moves left, which ensures optimality for certain search algorithms such as A*.[1]

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    Quote Originally Posted by SLD View Post
    I don’t understand how permutations can be even or odd. So take the group S4, which has 24 permutations, 12 odd, and 12 even.

    OK. What is that? How do you determine whether a given permutation is odd or even?

    TIA!

    SLD
    Every permutation can be written as a product of transpositions. If the product is of an even number of transpositions, then it is an even permutation, and vice versa.

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    Quote Originally Posted by SLD View Post
    I don’t understand how permutations can be even or odd. So take the group S4, which has 24 permutations, 12 odd, and 12 even.

    OK. What is that? How do you determine whether a given permutation is odd or even?

    TIA!

    SLD
    1234 - even
    2134 - odd
    2143 - even
    2413 - odd
    .........


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