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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?

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up vote 3 down vote accepted

The papers and contain some information about statistics on generalizations of Lehmer codes called $s$-inversion sequences.

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@Richard Stanley: Thank you! – Rodrigo A. Pérez Feb 18 '13 at 17:50

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