$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.

Let $N^+$ denote the space of uni-upper-triangular matrices in $\SL(n,\mathbb{R})$, and $N^+_{\geq 0} \subseteq \SL_{\geq 0}(n,\mathbb{R})$ the totally nonnegative parts of these spaces.

There are two stratifications of $N^+_{\geq 0}$ that I want to compare.

The first is what I'll call the **Catalan stratification**. It's the stratification based on the location of nonzero entries for a matrix $M \in N^+_{\geq 0}$, which can easily be seen to necessarily lie below/to the left of a Dyck path. E.g., one stratum in the case $n=5$ is:
$$ \begin{pmatrix} 1 & * & * & 0 & 0 \\ 0 & 1 & * & * & 0 \\ 0 & 0 & 1 & * & 0 \\ 0 & 0 & 0 & 1 & * \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix} $$
where the $*$'s denote the nonzero entries. So there are $C_n = \frac{1}{n+1}\binom{2n}{n}$ many strata here, and the closure relation among these strata is just containment of Dyck paths.

The second is more sophisticated; I'll call it the **Bruhat stratification**. For $1 \leq i \leq n-1$, let $x_i(t)$ denote the matrix with $1$'s along the main diagonal, $t$ in the $i$th spot right above the main diagonal, and $0$'s elsewhere. It's a theorem of Lusztig that for any permutation $w \in S_n$ and any reduced word $(i_1,i_2,\dotsc,i_\ell)$ of $w$, the map $(t_1,\dotsc,t_\ell)\to x_{i_1}(t_1)\dotsb x_{i_\ell}(t_\ell)$ is a homeomorphism of $\mathbb{R}_{>0}^{\ell}$ onto $Y^o_w := B^{-1}wB^{-1} \cap N^+_{\geq 0}$, where $B^-$ are the lower-triangular matrices in $\SL(n,\mathbb{R})$. (See Hersh - Regular cell complexes in total positivity for background on this.) So we have a stratification of $N^+_{\geq 0}$ indexed by permutations in $S_n$, and the closure relations among these strata is the (strong) Bruhat order on these permutations.

**Question**: Is it true that every Catalan stratum is a union of Bruhat strata? If so, what's the resulting map from permutations to Dyck paths?

I suspect the map from permutations to Dyck paths might be something like the "Cambrian congruences" of Nathan Reading. But I also suspect that this question may have already been studied somewhere.

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