A Cluster Character with Coefficients for Cluster Category

We introduce a new cluster character with coefficients for category $\mathcal{C}$ and rather than using Frobenius $2$-Calabi-Yau realization to incorporate into the representation-theoretic model algebra, as done by Fu Keller, we exploit intrinsic properties of $\mathcal{C}$. For this purpose, define an ice quiver associated each tilting object in In Dynkin case $\mathbb{A}_n$, also prove that mutation class given direct sum all projective objects is bijection set quivers