Arrays of permutations

Inversion (discrete mathematics)

These are some examples of similar permutations ordered in arrays.

Each permutation is represented in four ways:

For the last permutation in each array the permutation matrix is shown on the right.

In place ${\displaystyle (i,j)}$ is the cycle ${\displaystyle (j,j+i)}$. E.g. in place ${\displaystyle (7,1)}$ is the cycle ${\displaystyle (1,8)}$.   (The array of cycles corrsponds to the transposed array of 2-element subsets.)

The left column are the permutations whose cycles are ${\displaystyle (1,n,...,3,2)}$. Their index numbers are A007489 = 0, 1, 3, 9, 33, 153, 873, 5913...

The left column are the permutations whose cycles are ${\displaystyle (1,2,3,...,n)}$. Their index numbers are A001563 = 0, 1, 4, 18, 96, 600, 4320, 35280...