What is the expected number of fixed points in permutations?

Thread starter Dragonfall
Start date Aug 19, 2008
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Fixed points Permutations Points

In summary, fixed points in permutations are elements that do not change position after a permutation has been applied. They can be calculated by comparing the original permutation with the resulting permutation and are significant in understanding the structure and properties of the permutation. A permutation can have multiple fixed points, with the maximum number being equal to its length. Fixed points are also related to cycles in permutations, with fixed points being equivalent to cycles of length 1 and the number of fixed points being equal to the number of cycles of length 1.

Aug 19, 2008

Dragonfall

What is the expected number of fixed points in permutations? I got 1 as answer.

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Aug 19, 2008

HallsofIvy

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I don't see any immediate way to prove it but that's what it looks like to me.

1. What are fixed points in permutations?

Fixed points in permutations refer to the elements in a permutation that remain in the same position after the permutation has been applied.

2. How are fixed points calculated in a permutation?

Fixed points can be calculated by comparing the original permutation with the resulting permutation after it has been applied. Any elements that are in the same position in both permutations are considered fixed points.

3. What is the significance of fixed points in permutations?

Fixed points in permutations are important because they provide information about the structure and properties of the permutation. They can also be used in various mathematical and scientific applications, such as in cryptography and group theory.

4. Can a permutation have more than one fixed point?

Yes, a permutation can have multiple fixed points. In fact, the maximum number of fixed points a permutation can have is equal to its length.

5. How are fixed points related to cycles in permutations?

Fixed points and cycles are closely related in permutations. A fixed point can be seen as a cycle with a length of 1, while cycles of length greater than 1 do not have any fixed points. In other words, the number of fixed points in a permutation is equal to the number of cycles of length 1.