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

Let $V(1)$ be the natural representation of $U(\mathfrak{sl}_2).$ 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 $U_q^{\mathrm{res}}(\mathfrak{sl}_2)$. 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.