Combinatorial insights on tensor power decomposition in $U_q({\mathfrak{s}\mathfrak{l}}_2)-$module

Let $V(1)$ be the natural representation of $U({\mathfrak{s}\mathfrak{l}}_2)\mathrm{.}$ The multiplicities of $V(k)$ in $V(1{)}^{\otimes N}$ have multiple interpretations in combinatorics. In this paper, we investigate one such combinatorial interpretation of their multiplicities. Furthermore, we extend our analysis to the two-dimensional representation $T(1)$ of the restricted quantum enveloping algebra $U_q^{\mathrm{r}\mathrm{e}\mathrm{s}}({\mathfrak{s}\mathfrak{l}}_2)$. By employing combinatorial techniques, this study aims to elucidate the multiplicity of $T(k)$ within $T(1{)}^{\otimes N}$ while also offering a comprehensive analysis of the asymptotic behavior of these multiplicities as the parameter $N$ approaches infinity.