# Arrays of permutations

 Inversion (discrete mathematics)  Permutation ${\mathit {373}}$ from the array below

These are some examples of similar permutations ordered in arrays.

Each permutation is represented by its:

• place-based inversion set
• Rothe diagram (including the matrix representation as dots)
• left-inversion vector (0s represented by dots, the leading 0s omitted)
• reverse colexicographic index, i.e. the left-inversion vector interpreted as a little-endian factorial number

For the last permutation in each array the corresponding permutation matrix is also shown. ${\mathit {6936}}$ $=(124875)(36)$  ${\mathit {20533}}$ $=(157842)(36)$  ${\mathit {23616}}$ $=(15)(26)(37)(48)$  ${\mathit {5167}}$ $=(12)(34)(56)(78)$  Transpositions     This array corrsponds to the inverted array of 2-element subsets: In place $(i,j)$ is the cycle $(j,j+i)$ , e.g. $(18)$ in place $(7,1)$ . ${\mathit {36153}}=(18)$  In place $(i,j)$ is the set $\{i,i+j\}$ . ${\mathit {40319}}$ $=(18)(27)(36)(45)$  ${\mathit {5913}}$ $=(18765432)$  ${\mathit {35280}}$ $=(12345678)$ Circular shifts to the left, i.e. permutations whose cycle notation is of the form $(1~n~...~3~2)$ : = 0, 1, 3, 9, 33, 153, 873, 5913... Circular shifts to the right, i.e. permutations whose cycle notation is of the form $(1~2~3~...~n)$ : = 0, 1, 4, 18, 96, 600, 4320, 35280...