Arrays of permutations

Inversion (discrete mathematics)

Triangle of possible inversions of 8-element permutations

These are some examples of similar permutations ordered in arrays.

Each permutation is represented in four ways:

inversion set (place-based)	Rothe diagram (red entries) and permutation matrix (black dots)
left inversion count (0s represented by dots, leading 0s omitted)	reverse colexicographic index (left inversion count interpreted as a reversed factorial number)

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

The A-numbers of the number triangles work as links.

#A211366: alternating parity	#A211365: separated by parity
#A211367: big transpositions	#A211368: small transpositions
#A211369: single transpositions	#A100630: concentric transpositions
#A211370: circular shifts to the left	#A051683: circular shifts to the right

alternating parity

inversion set and inversion vector of permutation $373$

A211366

$(1,5,7,8,4,2)(3,6)$

separated by parity

A211365

$(1,2,4,8,7,5)(3,6)$

big transpositions

A211367

$(1,5)(2,6)(3,7)(4,8)$

small transpositions

A211368

$(1,2)(3,4)(5,6)(7,8)$

single transpositions

array of 2-element subsets

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

A211369

$(1,8)$

concentric transpositions

A100630

$(1,8)(2,7)(3,6)(4,5)$

circular shifts to the left

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

A211370

$(1,8,7,6,5,4,3,2)$

circular shifts to the right

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

A051683

$(1,2,3,4,5,6,7,8)$