Brauer tree algebras have $\binom{2n}{n}$ $2$-tilting complexes

We show that any Brauer tree algebra has precisely $\binom{2n}{n}$ $2$-tilting complexes, where $n$ is the number of edges of the associated Brauer tree. More explicitly, for an external edge $e$ and an integer $j\neq0$, we show that the number of $2$-tilting complexes $T$ with $g_e(T)=j$ is $\binom{2n-|j|-1}{n-1}$, where $g_e(T)$ denotes the $e$-th of the $g$-vector of $T$. To prove this, we use a geometric model of Brauer graph algebras on the closed oriented marked surfaces and a classification of $2$-tilting complexes due to Adachi-Aihara-Chan.