
2
In One Line and Anywhere. Permutations as
Linear Orders. Inversions.
2.1 Inversions
2.1.1 The Generating Function of Permutations by Inver-
sions
In Section 1.3, we looked at descents of permutations. That is, we studied
instances in which an entry in a permutation was larger than the entry directly
following it. A more comprehensive permutation statistic is that of inversions.
This statistic will look for instances in which an entry of a permutation is
smaller than some entry following it (not necessarily directly).
DEFINITION 2.1 Let p = p
1
p
2
···p
n
be a permutation. We say that
(p
i
,p
j
) is an inversion of p if i<j but p
i
>p
j
.
Example 2.2
Permutation 31524 ...