5 inferring the permutation
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My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]


What the Cycles Represent

The following diagram shows the first cycle walk: {2, 14, 16, 20, 15, 8, 23, 21, 19, 7}. Note that the cycle forms a closed loop (position 7 connects to position 2).

cycle


Inferring the permutation from the cycles

Suppose we knew the original ordering and the cycles but didn't know the second ordering:

Permute[order1, Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 
25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]]

{"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", \ "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"}

% === order2

True

My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]


What the Cycles Represent

The following diagram shows the first cycle walk: {2, 14, 16, 20, 15, 8, 23, 21, 19, 7}. Note that the cycle forms a closed loop (position 7 connects to position 2).

cycle

My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]


What the Cycles Represent

The following diagram shows the first cycle walk: {2, 14, 16, 20, 15, 8, 23, 21, 19, 7}. Note that the cycle forms a closed loop (position 7 connects to position 2).

cycle


Inferring the permutation from the cycles

Suppose we knew the original ordering and the cycles but didn't know the second ordering:

Permute[order1, Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 
25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]]

{"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", \ "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"}

% === order2

True

4 cycle closed
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My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]


What the Cycles Represent

The following diagram shows the first cycle walk: {2, 14, 16, 20, 15, 8, 23, 21, 19, 7}. Note that the cycle forms a closed loop (position 7 connects to position 2).

enter image description herecycle

My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]


What the Cycles Represent

The following diagram shows the first cycle walk: {2, 14, 16, 20, 15, 8, 23, 21, 19, 7}

enter image description here

My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]


What the Cycles Represent

The following diagram shows the first cycle walk: {2, 14, 16, 20, 15, 8, 23, 21, 19, 7}. Note that the cycle forms a closed loop (position 7 connects to position 2).

cycle

3 explanation of cycles given
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My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]


What the Cycles Represent

The following diagram shows the first cycle walk: {2, 14, 16, 20, 15, 8, 23, 21, 19, 7}

enter image description here

My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]

My interpretation of your question is: What cycles describe the permutation from order1 to order2?

order1 = {"G", "O", "V", "Y", "C", "H", "P", "W", "I", "Q", "X", "J", "D", "K", "R", "Z", "L", "M", "S", "N", "A", "E", "T", "B", "F", "U"};
order2 = {"G", "P", "L", "Y", "X", "I", "S", "R", "H", "D", "C", "B", "M", "O", "N", "K", "Q", "F", "A", "Z", "T", "U", "W", "V", "E", "J"};

If that is indeed what you are asking, FindPermutation will give the cycles:

FindPermutation[order1, order2]

Cycles[{{2, 14, 16, 20, 15, 8, 23, 21, 19, 7}, {3, 24, 12, 26, 22, 25, 18, 13, 10, 17}, {5, 11}, {6, 9}}]


What the Cycles Represent

The following diagram shows the first cycle walk: {2, 14, 16, 20, 15, 8, 23, 21, 19, 7}

enter image description here

2 added 1 characters in body
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