Combinatorics of Permutations, 3rd Edition

1In One Line and Close. Permutations as Linear Orders.

DOI: 10.1201/9780429274107-1

1.1 Descents

The “most orderly” of all n-permutations is obviously the increasing permutation $123\cdots n$. 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 Definition of Descents

DEFINITION 1.1 Let $p=p_1{p}_2\cdots p_n$ 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 p, while 1, 3, 4, and 5 are ascents of p.  ▯

Note that the descents ...