A permutation can be described in matrix notation by a 2 by n matrix with the numbers 1, ..., n in the first row and the images of 1, 2, ..., n (in that order) in the second row. Since there are n! possibilities to fill the second row, the following theorem holds.

The first row of the 2 by n matrix describing a permutation g S_n, is always 1, 2, ..., n and hence yields no essential information. We will often omit the first row; the permutation is then given in list notation. For example, [[1, 2, 3], [3, 1, 2]] becomes [3, 1, 2] in list notation.
However, the matrix notation is useful for calculating products and inverses.

Product
To calculate gh for two permutations g, h S_n, we first look up, for each i X, the value h(i), then we look for this value in the first row of the g matrix; below this entry you find gh(i).

Inverse permutation
If g is written as the 2 by n matrix M, then the inverse of g is described by the matrix obtained from M by interchanging the two rows and sorting the columns in such a way that the first row is again 1, 2, ..., n.