Laplace, permutations, notation question Watch

Ewan
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#1
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Hi, ok so I'm a little confused as to what this notation actually means...

Theres a symbol that kind of looks like a Pi, with 1<=i<j<=n below it, i'll call it P (yea I have no idea what it is, its in my notes but theres no example of what it is or what its called...)

L(sigma) = P [sigma(j) - sigma(i)]

If n = 5 for example... and sigma = (1,3,4,5), how would I find L(sigma)

Basically I'm looking for an example of how to work out this kind of question, or a link to a page which explains it with examples.

Cheers
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Ewan
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The topic i'm referring to is just below

"We can now define the transposition σ to be even if N(σ) is an even number, and odd if N(σ) is odd. This coincides with the definition given earlier but it is now clear that every permutation is either even or odd.

An alternative proof uses the polynomial"

on this page
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nota bene
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(Original post by Ewan)
Hi, ok so I'm a little confused as to what this notation actually means...

Theres a symbol that kind of looks like a Pi, with 1<=i<j<=n below it, i'll call it P (yea I have no idea what it is, its in my notes but theres no example of what it is or what its called...)
It is actually a capital pi, used like capital sigma, but for products instead.

L(sigma) = P [sigma(j) - sigma(i)]

If n = 5 for example... and sigma = (1,3,4,5), how would I find L(sigma)

Basically I'm looking for an example of how to work out this kind of question, or a link to a page which explains it with examples.

Cheers
Okay, so writing this out you have that
L(\sigma)=(\sigma_5-\sigma_4)(\sigma_5-\sigma_3)(\sigma_5-\sigma_2)(\sigma_5-\sigma_1)(\sigma_4-\sigma_3)(\sigma_4-\sigma_2)(\sigma_4-\sigma_1)(\sigma_3-\sigma_2)(\sigma_3-\sigma_1)(\sigma_2-\sigma_1)

Now, I'm not completely sure what the permutation (1,3,4,5) would mean? I know relatively little about this as I haven't ever studied it, but to me it doesn't make sense. E.g. (1,3,2,5,4) would simply be (23)(45)

edit: Having read a bit (1,3,4,5) would be a disjoint cycle meaning (13)(14)(15) ?
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CKsuper1
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When ur finished Ewan, let me know whether u get 12 n -12 for Q4 ......thx
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Ewan
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(Original post by nota bene)
It is actually a capital pi, used like capital sigma, but for products instead.


Okay, so writing this out you have that
L(\sigma)=(\sigma_5-\sigma_4)(\sigma_5-\sigma_3)(\sigma_5-\sigma_2)(\sigma_5-\sigma_1)(\sigma_4-\sigma_3)(\sigma_4-\sigma_2)(\sigma_4-\sigma_1)(\sigma_3-\sigma_2)(\sigma_3-\sigma_1)(\sigma_2-\sigma_1)

Now, I'm not completely sure what the permutation (1,3,4,5) would mean? I know relatively little about this as I haven't ever studied it, but to me it doesn't make sense. E.g. (1,3,2,5,4) would simply be (23)(45)

edit: Having read a bit (1,3,4,5) would be a disjoint cycle meaning (13)(14)(15) ?
That was just an example I made up... maybe not sensible, the one on my problem sheet is sigma = 1,3,4

if that makes any better sense

edit: and n = 4

though I *think* I see what to do one sec
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Ewan
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Ok, I see what you did, yes thats what my tutor told me to do.. however the n value confused me. Say you have

(1,2,3) and n = 3 you would do

sigma(3) - sigma(2) = 1-3 = -2
sigma(3) - sigma(1) = 1-2 = -1
sigma(2) - sigma(1) = 3-2 = 1

so L(sigma) = 2

but what if n=4?
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nota bene
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(Original post by Ewan)
Ok, I see what you did, yes thats what my tutor told me to do.. however the n value confused me. Say you have

(1,2,3) and n = 3 you would do

sigma(3) - sigma(2) = 1-3 = -2
sigma(3) - sigma(1) = 1-2 = -1
sigma(2) - sigma(1) = 3-2 = 1

so L(sigma) = 2
Yes.

edit: you can't have n=4 if you only have a cycle working on a set of three elements (unless 4->4 is implied)


Second edit: If you have (1,3,4) working on the set {1,2,3,4} then that is the same as (13)(14) working from the left. Just use n=4 in the defintion now and you get a product, I'll see what I get...

I make it
L(sigma)=12
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Ewan
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So how do I answer my question...

For n = 4 calculate L(sigma) when sigma = (1,3,4)

This is the only thing that gets me, otherwise its easy
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CKsuper1
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btw, does sigma = (1 3 4) and n=4 mean:

(Let @ = sigma)

@(1)=3
@(2)=2
@(3)=4
@(4)=1

(this is how I dne it)
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Ewan
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Cheers nota, I got it

(Original post by CKsuper1)
When ur finished Ewan, let me know whether u get 12 n -12 for Q4 ......thx
Yes, 12 and -12

EDIT: I did it like this:

(134) means (14)(13)

ie

1234
3214
3241

thus

(1-4)(1-2)(1-3)(4-2).... etc
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CKsuper1
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:thumbsup: Gr8! same method lol
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Ewan
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I still don't get analysis.... its like staring at a brick wall :p:
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