Palindromic Length in Free Groups: Reflections, Noncrossing Matchings, and Catalan Forms

Let $F=F(X)$ be a free group of finite rank, with palindromic length taken with respect to the fixed basis $X$. We embed $F$ as the index-two subgroup of the universal Coxeter group $W=F\rtimes_θ\langle t\mid t^2=1\rangle$, where $θ(x)=x^{-1}$ for $x\in X$, and prove $\mathrm{pl}(g)=\min{\ell_T(g),\