Graphical to Mathematical representation of changing the order of some elements

• arsenal_51
In summary, the conversation discussed a method for finding the necessary permutations to transform an ordered set of numbers to a specified order. This can be done graphically or mathematically using swappings. One method requires 2N-1 swappings, while the other more efficient method only requires N-1 swappings. Further research is needed to learn more about the efficient method.
arsenal_51
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: C123
and we have some operations: O12, O13 and O23
each of these operations act on those numbers changing their positions.

For example O12 will change the position of the first and second elements.
So let's say: O12 . C123 will equal: C213

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: C123
Target: C231

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: O12 . O23 . C123 = C231

One last example:
The container doesn't need to hold all of the numbers of the three number-space
Original: C12
Target: C31

Or: O23 . O12 . C12 = C31

So graphically its easy to find out the operations of any N number-space.
But how do we express that in a mathematical general expression?

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

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: O1i . O2j . C12 = Cij
But for a N number group instead of just a this small example that might not even be correct.

Last edited:
arsenal_51 said:

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: O1i . O2j . C12 = Cij
But for a N number group instead of just a this small example that might not even be correct.

Your notation is somewhat unconventional, but I think you mean the following: Given an ordered set of the first N counting numbers, i.e. ##S = (s_1\,s_2\,s_3\,\ldots\,s_N)##, you want to find ##X## "swappings" of the form ##(f_1\,t_1),\ldots,(f_X\,t_X)## such that their product will take the ordered set ##(1\,2\,3\,\ldots\,N)## to ##S##, i.e.
$$(1\,2\,3\,\ldots\,N)\cdot(f_1\,t_1)\cdot\ldots \cdot (f_X\,t_X)=(s_1\,s_2\,s_3\,\ldots\,i_N).$$
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_{N-1}## into position ##N-1## etc. till you end up with the ordered set ##S## you wanted.

This way you may need ##X=2N-1## 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 ##N-1## swappings, but not as easily as by my method (one element at a time goes into position ##1## and then to its proper place).

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.

1. How do you represent the change in order of elements graphically?

The change in order of elements can be represented graphically by using a bar graph, where the height of each bar represents the frequency or number of elements in that particular order.

2. What is the purpose of representing this change in a mathematical form?

Representing the change in a mathematical form allows for a more precise and quantitative analysis of the data. It also allows for easier comparison and identification of patterns or trends.

3. How do you convert the graphical representation to a mathematical one?

The graphical representation can be converted to a mathematical one by assigning numerical values to each bar on the graph and then using mathematical operations such as addition, subtraction, and multiplication to represent the change in order.

4. Can you provide an example of a graphical to mathematical representation of changing element order?

Sure, let's say we have a bar graph representing the order of letters in a word. The first bar represents the letter 'A', the second bar represents the letter 'B', and so on. We can convert this to a mathematical representation by assigning the letter 'A' a value of 1, 'B' a value of 2, and so on. If the order of the letters changes from 'ABC' to 'CAB', the mathematical representation would be (3+1+2) - (1+2+3) = -1, indicating a change in order.

5. How can the graphical to mathematical representation of changing element order be useful in scientific research?

This representation can be useful in various fields of science such as biology, chemistry, and physics, where changes in the order of elements can provide valuable insights. For example, in biology, it can be used to analyze changes in DNA sequences, and in chemistry, it can be used to study the reactivity of different elements in a compound.

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