### 48.1 Package combinatorics

The `combinatorics` package provides several functions to work with permutations and to permute elements of a list. The permutations of degree n are all the n! possible orderings of the first n positive integers, 1, 2, …, n. The functions in this packages expect a permutation to be represented by a list of those integers.

Cycles are represented as a list of two or more integers i_1, i_2, …, i_m, all different. Such a list represents a permutation where the integer i_2 appears in the i_1th position, the integer i_3 appears in the i_2th position and so on, until the integer i_1, which appears in the i_mth position.

For instance, [4, 2, 1, 3] is one of the 24 permutations of degree four, which can also be represented by the cycle [1, 4, 3]. The functions where cycles are used to represent permutations also require the order of the permutation to avoid ambiguity. For instance, the same cycle [1, 4, 3] could refer to the permutation of order 6: [4, 2, 1, 3, 5, 6]. A product of cycles must be represented by a list of cycles; the cycles at the end of the list are applied first. For example, [[2, 4], [1, 3, 6, 5]] is equivalent to the permutation [3, 4, 6, 2, 1, 5].

A cycle can be written in several ways. for instance, [1, 3, 6, 5], [3, 6, 5, 1] and [6, 5, 1, 3] are all equivalent. The canonical form used in the package is the one that places the lowest index in the first place. A cycle with only two indices is also called a transposition and if the two indices are consecutive, it is called an adjacent transposition.

To run an interactive tutorial, use the command `demo (combinatorics)`. Since this is an additional package, it must be loaded with the command `load("combinatorics")`.

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