Descent of a permutation ($\mathrm{Des}(a),\mathrm{des}(a)$)

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.

The major index of a permutation $\pi ={\pi }_1\cdots {\pi }_n$ is the sum of its permutation descent positions:It uses positions, rather than the values being compared. For $3142$, the permutation descents occur at positions one and three, giving $\mathrm{maj}=4$, whereas the descent number is two.

The major index has the same distribution over all permutations of $n$ elements as the inversion number:For $n=3$, the six values give $1+2q+2q^2+q^3$. Individual statistics need not agree: $312$ has major index one and two inversions. The shared generating polynomial is a statement about their distribution across the whole symmetric group.^[ref]