A permutation cycle is a mathematical concept that represents the rearrangement of a set of elements, where each element is moved to a different position in the set. It is often denoted by a set of parentheses, with the elements inside representing the order of the rearrangement.
Permutation cycles are typically written using cycle notation, which is a compact and efficient way of representing a permutation. In this notation, the elements that are moved are listed inside parentheses, with the first element being moved to the position of the second element, the second element to the position of the third element, and so on.
The composition of permutation cycles refers to combining two or more permutation cycles to form a single permutation. This is done by performing each cycle in order, starting with the rightmost cycle and moving towards the left. The resulting permutation is equal to the final position of each element after all cycles have been performed.
The composition of permutation cycles can be calculated by multiplying the cycles together, with the rightmost cycle being performed first. For example, if we have the cycles (1 2 3) and (2 3 4), their composition would be (1 2 3)(2 3 4) = (1 3 4).
The order of a permutation cycle is the number of elements that are moved in the cycle. This can also be thought of as the length of the cycle. For example, the cycle (1 2 3) has an order of 3, since it moves 3 elements in the set.