Define permutation10/2/2023 ![]() ![]() If you have a permutation that is the product of disjoint cycles: say three cycles, corresponding to lengths $n_1, n_2, n_3$, then the number of transpositions representing this permutation can be computed by the parity of $(n_1 - 1)+(n_2 - 1) + (n_3 - 1)$ or simply the parity (oddness/evenness) of $n_1+n_2+n_3 - 1$ So a cycle with a length that is even (has an even number of elements) is ODD, and a cycle with a length that is odd (has an odd number of elements) is EVEN. You'll see that the number of transpositions in a product corresponding to a permutation that is a cycle of length $n$ can be expressed as the product of $n - 1$ transpositions. One can always resort to following the pattern: If you know cycle notation, knowing the parity (oddness/evenness) can be found fairly easily. There are many ways to write a permutation as the product of transpositions, and they can vary in length, but those products will have either an odd or an even number of factors, never both. an even number of transpositions $\iff$ even permutation.an odd number of transpositions $\iff$ odd permutation.Every permutation can be expressed as the product of one and only one of the following:
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