Permutation indices
Here’s just a quick post of a plot I did a quite a while back:
This animation shows for 
through to 
the number of permutations with a given number of inversions and major index. I love the way that discrete structure slowly falls away into something nearly continuous. I would have included more frames in the animation, but as there are 
permutations in 
it gets expensive to compute these by brute force counting. I would need to come up with a more effective method.
An inversion of a permutation 
on 
is a pair of elements 
with 
such that 
That is, the permutation reverses their order. The inversion number, 
is the number of such pairs that a given permutation has. For example, the permutation 
(regarded as a word, not a cycle) has inversions 
and so 
A descent in a permutation is a position 
with 
where a larger number occurs immediately before a smaller one. That is, 
The major index, 
is defined as the sum of the descents. For example, in the permutation we have descents at the first position (3 falls to 1), and at the third (4 falls to 2). So the major index of this permutation is 
The plot shows at position the number of permutations with exactly inversions, and major index 
as a normalised shade of grey, darker meaning larger. Indices increase left to right and top to bottom.
You can easily see from the symmetry of the plots that permutations are equidistributed with respect to these two indices, so that we can expect the polynomial where the coefficient of 
is the number of permutations with inversion number and major index will be a symmetric polynomial in 
and 