Let \(V=\{a_i,~i\in I\}\) be a finite (or infinite) set with a given index \(I\) of its elements \(I=\{1,\ldots,n,\ldots\}\). A permutation of \(V\) is a bijective map on this order \(\pi:I\mapsto I\).
A permutation can be generated by selecting elements from an ordered collection \(A\) and moving them from this collection into a new ordered collection \(P\).
There are different ways a permutation \(\pi\) can be notated. As a permutation does not depend on the specific elements of \(a_i\in V\), all notations are related to the index set \(i\).
Possibility 1 (Two-Line Notation): \[\pi=\left( \begin{matrix} 1 & 2 & 3 & \cdots & n-1 & n \\ \pi(1) & \pi(2) & \pi(3) & \cdots & \pi(n-1) & \pi(n) \\ \end{matrix} \right)~~~~~~~~~~~~\text{e.g. for }n=9:~~~~\left( \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 7 & 8 & 6 & 4 & 9 & 3 & 2 & 1 & 5\\ \end{matrix} \right)\] As the indices of the first line always appear in their natural order, they do not provide any additional information. Therefore the following one-line notation is more common:
Possibility 2 (One-Line Notation): \[\pi=( \begin{matrix} \pi(1) & \pi(2) & \pi(3) & \cdots & \pi(n-1) & \pi(n) \\ \end{matrix} )~~~~~~~~~~~~\text{e.g. for }n=9:~~~~( \begin{matrix} 7 & 8 & 6 & 4 & 9 & 3 & 2 & 1 & 5\\ \end{matrix} )\]
Possibility 3 (Cycle Notation): The third possibility focuses on the effect of successively applying the permutation. In this context, a permutation is equivalent to a set of cycles, i.e. ordered partitions \(C\) on the set \(I\) such that the permutation \(\pi\) does not changes the order and composition of the partitions. In the previous example $[1,7,2,8],$ $[3,6],$ $[4 ],$ $[5,9]$ are cycles, so \[\pi=\{C_1, C_2, \ldots, C_k\}~~~~~~~~~~~~\text{e.g. for }n=9:~~~~\{[1,7,2,8], [3,6], [^4], [5,9]\}\]
Corollaries: 1
Definitions: 2 3 4
Explanations: 5
Parts: 6
Problems: 7
Proofs: 8 9 10
Propositions: 11 12 13
Solutions: 14