Arrays of permutations

| Inversion (discrete mathematics)

Triangle of possible inversions of 8-element permutations | Permutation  373 {\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.

A211365

|  6936 {\displaystyle {\mathit {6936}}}
= ( 124875 ) ( 36 ) {\displaystyle =(124875)(36)} | A211366 |  20533 {\displaystyle {\mathit {20533}}}
= ( 157842 ) ( 36 ) {\displaystyle =(157842)(36)}

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A211367     Chains of transpositions

|  23616 {\displaystyle {\mathit {23616}}}
= ( 15 ) ( 26 ) ( 37 ) ( 48 ) {\displaystyle =(15)(26)(37)(48)} | A211368     Rows of transpositions |  5167 {\displaystyle {\mathit {5167}}}
= ( 12 ) ( 34 ) ( 56 ) ( 78 ) {\displaystyle =(12)(34)(56)(78)}

A211369     Transpositions     This array corrsponds to the inverted array of 2-element subsets: In place  ( i , j ) {\displaystyle (i,j)} is the cycle  ( j , j + i ) {\displaystyle (j,j+i)} , e.g.  ( 18 ) {\displaystyle (18)} in place  ( 7 , 1 ) {\displaystyle (7,1)} .

|  36153 = ( 18 ) {\displaystyle {\mathit {36153}}=(18)}
In place  ( i , j ) {\displaystyle (i,j)} is the set  { i , i + j } {\displaystyle \\{i,i+j\\}} . | A100630     Nested transpositions |  40319 {\displaystyle {\mathit {40319}}}
= ( 18 ) ( 27 ) ( 36 ) ( 45 ) {\displaystyle =(18)(27)(36)(45)}

A211370     Circular shifts to the left in an interval

|  5913 {\displaystyle {\mathit {5913}}}
= ( 18765432 ) {\displaystyle =(18765432)} | A051683     Circular shifts to the right in an interval |  35280 {\displaystyle {\mathit {35280}}}
= ( 12345678 ) {\displaystyle =(12345678)}

Circular shifts to the left,
i.e. permutations whose cycle notation is of the form  ( 1   n   . . .   3   2 ) {\displaystyle (1~n~...~3~2)} :
A007489 = 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 ) {\displaystyle (1~2~3~...~n)} :
A001563 = 0, 1, 4, 18, 96, 600, 4320, 35280...