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Permutations as a Product of Transpositions As in the shuttle puzzlethe applet below allows you to connect any two poles with vertical "shuttles". Each such shuttle defines a transposition, i. Several subsequent shuttles counted from top to bottom define a permutation that "follows the shuttles in the following manner: For every pole in turn, start from the top, slide down till the first intersection with a shuttle, if any, to the end of the shuttle, from there again down to the first intersection with another shuttle, then follow this shuttle to the end, then down again, and so on.

When eventually you reach a pole's bottom, this pole's number corresponds to the number of the pole where the descent has started.

As we knowsuch a procedure indeed defines a permutation. This permutation is naturally treated as a product of successive transpositions.

What if applet does not run? The representation of a permutation as a product of transpositions is not unique, but the parity of the number of transpositions in the product is a feature of the permutation and does not depend on the representation.

Any permutation can be represented as a product of cycles.

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Every cycle is shown to be a product of transpositions. Thus, every permutation can be represented as a product of transpositions.

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In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd benjaminpohle.com any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of.

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Permutations as a Product of Transpositions