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Graphical to Mathematical representation of changing the order of some elements 
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#1
Dec3012, 06:14 PM

P: 3

I have a question that is a little hard to explain, since i don't know the name of this method, but I'll try my best, if anyone knows the name please do tell me.
So let's say we have three numbers, 1 2 3 (in this order) and we have a container for this numbers: C_{123} and we have some operations: O_{12}, O_{13} and O_{23} each of these operations act on those numbers changing their positions. For example O_{12} will change the position of the first and second elements. So lets say: O_{12} . C_{123} will equal: C_{213} And if we want to find out what operations to use when we have the original Container and the target Container we can do it easily graphically. For example: Original: C_{123} Target: C_{231} This can be done graphically: The point where the lines intercept represent the operation between those two numbers. And the order is important, since these operations are not commutable. So that's the same as: O_{12} . O_{23} . C_{123} = C_{231} One last example: The container doesn't need to hold all of the numbers of the three numberspace Original: C_{12} Target: C_{31} Or: O_{23} . O_{12} . C_{12} = C_{31} So graphically its easy to find out the operations of any N numberspace. But how do we express that in a mathematical general expression? 


#2
Dec3012, 06:19 PM

P: 772

I'm not exactly sure what you're asking. These are just basic permutations, so cycle notation should communicate everything that you need.



#3
Dec3012, 06:43 PM

P: 3

Thanks for your reply, i'm going to read about that.
I'm trying to find out the permutations needed to do mathematically for any N number group, knowing only the original and the final state. Ideally something of the format: O_{1i} . O_{2j} . C_{12} = C_{ij} But for a N number group instead of just a this small example that might not even be correct. 


#4
Dec3012, 08:23 PM

P: 181

Graphical to Mathematical representation of changing the order of some elements
$$ (1\,2\,3\,\ldots\,N)\cdot(f_1\,t_1)\cdot\ldots \cdot (f_X\,t_X)=(s_1\,s_2\,s_3\,\ldots\,i_N). $$ This is possible, and you can construct the pairs ##(f_k\,t_k)## quite easily. I'll just hint by saying this much: choose ##(f_1\,t_1)## so that it swaps the elements ##1## and ##s_N##, and then let ##(f_2\,t_2)=(1\,N)##. This means that those two swapping put ##s_N## at position ##N##. In the next step you place ##s_{N1}## into position ##N1## etc. till you end up with the ordered set ##S## you wanted. This way you may need ##X=2N1## swappings, and it's actually possible to get from ##(1\,2\,3\,\ldots\,N)## to any ##(s_1\,s_2\,s_3\,\ldots\,s_N)## with only ##N1## swappings, but not as easily as by my method (one element at a time goes into position ##1## and then to its proper place). 


#5
Dec3112, 02:22 PM

P: 3

Thank you, i will try out your method, it seems pretty clear.
After i try that out i would like to check that other more efficient method you were talking about where you only need N  1 swapping operations. Do you know where i can read more about that other method or the name of it? Thanks again. 


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