= Descent of a permutation
{title2=$\operatorname{Des}(a),\ \operatorname{des}(a)$}
= Permutation descent
{synonym}
A descent of a <permutation> written in one-line form $a_1\cdots a_n$ is a position $i\in\{1,\ldots,n-1\}$ with $a_i>a_{i+1}$. Its descent set and number are
$$
\operatorname{Des}(a)=\{i:a_i>a_{i+1}\},\qquad\operatorname{des}(a)=|\operatorname{Des}(a)|.
$$
These are adjacent comparisons, unlike <inversions of a permutation>, which compare arbitrary ordered pairs of positions.
For $3142$, the descents occur at positions one and three, since $3>1$ and $4>2$. The middle comparison is increasing. The identity permutation has no descents, while the reversed order $n\cdots21$ has $n-1$.
The <Eulerian number> $A(n,k)$ counts permutations with $k$ descents under the stated indexing convention. The <major index> instead sums their positions; for $3142$ it is $1+3=4$, illustrating a different statistic derived from the same descent set.
Back to article page
