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 $\mathcal C_{-d}(H)$ and silting objects of ${\mathcal {D}}^{\mathrm {b}}(H)$ contained in ${\mathcal {D}}^{\le 0}\cap {\mathcal {D}}^{\ge 1-d}$ for hereditary algebra $H$ of Dynkin type and $d\ge 1$. We show that the number of $d$-SMSs in $\mathcal C_{-d}(H)$ 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.