
1
In One Line and Close. Permutations as
Linear Orders.
1.1 Descents
The “most orderly” of all n-permutations is obviously the increasing permuta-
tion 123 ···n. All other permutations have at least some “disorder” in them;
for instance, it happens that an entry is immediately followed by a smaller
entry in them. This simple phenomenon is at the center of our attention in
this Section.
1.1.1 The Definition of Descents
DEFINITION 1.1 Let p = p
1
p
2
···p
n
be a permutation, and let i<n
be a positive integer. We say that i is a descent of p if p
i
>p
i+1
. Similarly,
we say that i is an ascent of p if p
i
<p
i+1
.
Example 1.2
Let p = 3412576. Then 2 and 6 are descents of ...