Exceptional sequences of type $B_n\mathrm{/}{C}_n$ and those in the abelian tube

We examine clusters in the cluster tube of rank $n+1$ using exceptional sequences in the abelian tube of rank $n+1$. Although the abelian tube has more exceptional sequences than the module categories of type $B_n\mathrm{/}{C}_n$, we obtain a bijection between the set of signed exceptional sequences of any length in these categories. This bijection gives a reinterpretation of the formula of Buan-Marsh-Vatne comparing clusters of type $B_n\mathrm{/}{C}_n$ with maximal rigid objects in the cluster tube of rank $n+1$. The bijection passes through the set of “augmented” rooted labeled trees.