I am looking at words $\alpha_1 \ldots \alpha_n$, where $\alpha_j \in \{ 1, \ldots, j \}$. Thinking of $\alpha_j$ as a height, these words can be interpreted as left-to-right paths on the positive square lattice which stay under the diagonal. They can also be thought as a Lehmer code for permutations. Obviously, there are $n!$ such words.

Assign to each word $w$ the number $\text{maj}(w)$ equal to the sum of indices $j$ such that $\alpha_j > \alpha_{j+1}$. This is the *major index* of $w$. Unlike the major index on the set of permutations, the major index on the set of paths does not produce a symmetric distribution.

Do you know any references that discuss this (or any other statistic)

on paths?