Positive Fuss-Catalan numbers and Simple-minded systems in negative Calabi-Yau categories

We establish a bijection between $d$-simple-minded systems ($d$-SMSs) of $(-d)$-Calabi-Yau cluster category ${\cal C_{-d}}(H)$ and silting objects of ${\cal D^{\rm b}}(H)$ contained in $\cal D^{\le 0}\cap \cal D^{\ge 1-d}$ for hereditary algebra $H$ of Dynkin type and $d\ge 1$. We show that the number of $d$-SMSs in ${\cal C_{-d}}(H)$ is the positive Fuss-Catalan number $C_{d}^{+}(W)$ of the corresponding Weyl group $W$, by applying this bijection and Buan-Reiten-Thomas' and Zhu's results on Fomin-Reading's generalized cluster complexes. Our results are based on a refined version of silting-$t$-structure correspondence.