Lehmer code

For a permutation $p_0,\dots ,p_{n-1}$ of $n$ ordered distinct values, its Lehmer code has entries $c_i=\#\{j>i:p_j<p_i\}$. Its zero-based lexicographic order rank is ${\sum }_{i=0}^{n-1}{c}_i(n-1-i)!$. Each entry chooses among the still unused values; the factorial weights count all completions with a smaller choice at the first differing position.