# Arrays of permutations

 Triangle of possible inversions of 8-element permutations Permutation ${\displaystyle {\mathit {373}}}$ 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.

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