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Let $V(1)$ be the natural representation of $U(\mathfrak{sl}_2).$ Themultiplicities of $V(k)$ in $V(1)^{\otimes N}$ have multiple interpretations incombinatorics. In this paper, we investigate one such combinatorialinterpretation of their multiplicities. Furthermore, we extend our analysis tothe two-dimensional representation $T(1)$ of the restricted quantum envelopingalgebra $U_q^{\mathrm{res}}(\mathfrak{sl}_2)$. By employing combinatorialtechniques, this study aims to elucidate the multiplicity of $T(k)$ within$T(1)^{\otimes N}$ while also offering a comprehensive analysis of theasymptotic behavior of these multiplicities as the parameter $N$ approachesinfinity.

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