Cycle notation of permutations

[gottfried]

Homework Statement

How many elements of each cycle-type are there in S₅?

The Attempt at a Solution

One way of working this out would be to write out each permutation and see how many 2-cycles, 3-cycles ,4-cycles and 5-cycles there are but given that there are 5! permutations this would take very long.

Does anybody know of an easier way to see the answer?

 

[I like Serena]

Hi gottfried!

Rather than writing them all out, I suggest only starting to write them all out.
And while doing so, trying to see patterns that be generalized, so you don't have to write them all out.

Perhaps more to the point.
Suppose you want to know how many 3 cycles there are.
How many choices do you have for each number, considering they have to be different?
That is, how many 3-tuples are there with 3 different numbers?

And how many of those 3-tuples are duplicates as 3-cycles?

Btw, there are more cycle-types than the ones you mention.
So don't forget to check you get a total of exactly 5!=120 cycle types.

 

[gottfried]

Hi
Thanks for the help.

So after considering what you have said I can see that any cycle written (abc) is the same as (bca) and (cab). This means that while counting the number of 3-tuples it will give you an answer 3 times the number of 3-cycles. Same logic holds for 2, 4 and 5 cycles.
Therefore there are
(5.4)/2 = 10:# 2-cycles.
(5.4.3)/3 = 20:#3-cycles
(5.4.3.2)/4=30:#4-cycles
(5.4.3.2)/5=24:#5-cycles
also the identity (12)(12) so that is 85 cycles.

This part makes total sense (assuming I'm right) but I don't know how to count the number of cycles in the form (ab)(cd) and (ab)(cde)

 

[Robert1986]

Well, first, I'd write out what the different cycle types there is. As you probably know, there are p(5) (the partition function) different cycle types. I'd start there, and for each cycle type, do as I Like Serena said and see what you come up with. However, I Like Serena said to make sure you come up with 5! cycle types, but I think what he meant was make sure you come up with 5! different permutations. As I said, there are only p(5) = 7 cycle types.

 

[I like Serena]

Good!

And yes, as Robert said, I meant that there should be 120 different permutations.

So for (ab)(cde), how many 5-tuples of 5 different numbers?
Is (ba)(cde) the same permutation?
And (ba)(ced)?
In other words, how many duplications?

The form (ab)(cd) is actually the most complex form.
But let's do (ab)(cde) first.

Btw, the so called "cycle type" of (ab)(cde) is (3,2), which should not be confused with (3 2) which is a 2-cycle.