Verifying a Permutation: Even or Odd?

In summary, the conversation discusses the process of determining whether a given permutation is even or odd. The correct method is to factor it into disjoint cycles and count the number of odd permutations. If the count is even, the permutation is even, and if the count is odd, the permutation is odd. The terminology used initially caused some confusion, but it was clarified that an "even cycle" refers to an "odd number of transpositions." The suggested formulation removes ambiguity and points out that the decomposition into disjoint cycles is unique except for order.
  • #1
catherinenanc
49
0
1. I think my professor gave me credit for this problem when he shouldn't have! I just want to understand for the test, so I want someone to verify for me.
2. How do you find out whether a given permutation is even or odd without factoring it into transpositions?
3. My answer was:
Factor it into disjoint cycles (not necessarily 2-cyles, or transpositions).
Count the number of even cycles (those with an odd r).
If this count is even, the permutation is even. If this count is odd, then the permutation is odd.

but in looking back over my work to study for the test, I think it should have been:
Factor it into disjoint cycles (not necessarily 2-cyles, or transpositions).
Count the number of odd cycles (those with an even r).
If this count is even, the permutation is even. If this count is odd, then the permutation is odd.

Am I right in doubting my previous answer?
 
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  • #2


Hi catherinenanc! :smile:

Nope. Your previous answer is correct.
An even cycle is an odd permutation.
With odd and odd you can make even, but with even and even you can only make even.
 
  • #3


I like Serena said:
Hi catherinenanc! :smile:

Nope. Your previous answer is correct.
An even cycle is an odd permutation.
With odd and odd you can make even, but with even and even you can only make even.

You mean to say that that an "even cycle" is an "odd number of transpositions", right?
(BTW, odd r means (x1,x2,x3,...,xr) like (2,3,5) or (1,5,6,8,9).)

Also, I agree that "With odd and odd you can make even, but with even and even you can only make even" - that's precisely why I think my answer was wrong.
For example, Take (abc)(def)(ghij)(klmn) and (abc)(def)(ghij)(klmn)(opqs).
The first would be even+even+odd+odd=even
The second would be even+even+odd+odd+odd=odd.
 
  • #4


catherinenanc said:
an "even cycle" is an "odd number of transpositions"

Or, rather, that an "even cycle" can be expressed as an "odd number of transpositions"
 
  • #6


Ok the problem here is that our terminology is off. I used a term in my answer "even cycle" that threw us off. My bad.
My point is, shouldn't this be correct instead of my original?:

Factor it into disjoint cycles (not necessarily 2-cyles, or transpositions).
Count the number of odd permutations (those with an even r=number of values shown in a row).
If this count is even, the permutation is even. If this count is odd, then the permutation is odd.
 
  • #7


Formulating it like that removes the ambiguity, which is better. :)

Btw, note that the decomposition into disjoint cycles is unique except for order.
 

Related to Verifying a Permutation: Even or Odd?

1. What is a permutation?

A permutation is a rearrangement of a set of elements. In other words, it is a way of organizing or ordering a group of items.

2. How do you determine if a permutation is even or odd?

To determine if a permutation is even or odd, you can use the parity rule. This rule states that if a permutation can be achieved by an even number of swaps, it is considered even. If it requires an odd number of swaps, it is considered odd.

3. Why is verifying a permutation important?

Verifying a permutation is important because it allows us to check if a given set of elements has been correctly rearranged according to a specific rule or method. It helps to ensure accuracy and avoid mistakes in data manipulation or analysis.

4. Can a permutation be both even and odd?

No, a permutation can only be either even or odd. This is because the parity rule states that a permutation can either be achieved by an even or odd number of swaps, not both.

5. Are there any real-life applications for verifying permutations?

Yes, verifying permutations has many real-life applications. For example, it is used in computer programming, cryptography, and data analysis to ensure that data is correctly sorted or organized. It is also used in solving mathematical problems and puzzles.

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