I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. The function $\ell$ gives the classical Coxeter length on $G$, that is $\ell(w)$ is the length of a shortest word in $S$ that yields $w$. The said reduced length is then defined as $$\ell_R(w) := \min \{r \mid w = t_1t_2\ldots t_r, \,\, t_i \in T, \,\, \ell(w) = \sum_{i=1}^r \ell(t_i)\}$$ and the authors show that the depth statistic discussed in the given paper satisfies $$dp = \frac{\ell + \ell_R}{2}$$ in many cases of interest.

I tried to find more information on $\ell_R$, but couldn't find the term in any other work. Has anyone already heard of the term and knows where to look?

What I am most interested in is the joint distribution of length and reduced length or length and depth (by the above formula, both problems are equivalent), that is the polynomials $$\sum_{g \in G} x^{\ell(g)}y^{\ell_R(g)} \text{ and } \sum_{g \in G} x^{\ell(g)}y^{dp(g)}$$ where we assume $G$ to be finite. I also couldn't find any paper discussing the length with either one of these functions together. While I would love to find answers for all finite Coxeter groups, it would already be great to get information on the special case $G = S_n$.

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