This page uses the published inversion number and the shifted Eulerian indexing $A(n,k)$

This page keeps the published inversion convention of Inversions, inversion number, the sign $\mathrm{sgn}(\sigma )=(-1{)}^{\mathrm{inv}(\sigma )}$, and even and odd permutations exactly as written: permutations act on $n=\{0,1,\dots ,n-1\}$, and an inversion is a pair $(i,j)$ with $i<j<n$ and $\sigma (i)>\sigma (j)$. Nothing here reverses that orientation or silently shifts to a $1,\dots ,n$ indexing set.

The same zero-based convention is used for descents and excedances, but the major index is still the classical sum of descent positions counted from $1$. Thus a descent at the zero-based position $i$ contributes $i+1$ to $\mathrm{maj}(\sigma )$.

The Eulerian number $A(n,k)$ counts permutations of $S_n$ with exactly $k$ descents, so $k$ ranges from $0$ to $n-1$ when $n\ge 1$. We also set $A(0,0)=1$, and the Eulerian polynomial is $A_n(t)={\sum }_{k=0}^{n-1}A(n,k)t^k$ for $n\ge 1$ with $A_0(t)=1$.