Another way of
describing permutations 
is the list notation,
where we put the list
of their values in square brackets:
In order to visualize permutations we can use a board consisting of boxes in the
second quadrant, where we put a black button 
into the box with the pair of coordinates
if and only 
obtaining the Rothe diagram
of 
This diagram was introduced by
H. A. Rothe already in 1800. For example
is equal to
The list notation gives rise to the Lehmer code
which we are going to describe next. In order to do this we consider
the set 
of inversions of 
:
An easy check (exercise 
) shows that the following holds:
.
Lemma
If we represent the inversion 
of by a white button
at 
then we fill exactly the boxes of the Rothe diagram
of where the black button of the row is in a column to the right and where
the black button of the column is lower down.
The resulting diagram 
is called the Lehmer diagram
of . For example, the superposition of the Rothe diagram and the Lehmer
diagram of 
is
There is a program, which calculates the number of inversions of a given permutation.
Exercises
.