 Descent of a permutation

A descent of a permutation written in one-line form $a_1\cdots a_n$ is a position $i\in \{1,\dots ,n-1\}$ with $a_i>a_{i+1}$. Its descent set and number areThese 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 $n\cdots 21$ 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.