Rooted labeled trees and exceptional sequences of type $B_n/C_n$

Abstract: We show that exceptional sequences in the abelian tube of rank $n$, which we denote $\mathscr{ W}n$, are related to exceptional sequences of type $B_n$ and $C_n$ and to those of type $B{n-1}$ and $C_{n-1}$. $\mathscr{W}n$ has $n^{n-1}$ exceptional sequences. These are in $1$-to-$n$ correspondence from the $n^n$ "augmented" rooted labeled trees with $n$ vertices which, in turn, are in bijection with exceptional sequences of type $B_n$ and $C_n$. By determining the probability distribution of relative projectives in these exceptional sequences, we show there are $(2n)!/n!$ signed exceptional sequences in $\mathscr{W}_n$ and we show that these are in bijection with signed exceptional sequences of type $B{n-1}$ and $C_{n-1}$ by combining the results of Buan-Marsh-Vatne and Igusa-Todorov.