Unified monogamy relations for the generalized $W$-class states beyond qubits

The monogamy of entanglement stands as an indispensable feature within multipartite quantum systems. We study monogamy relations with respect to any partitions for the generalized $W$-class (GW) states based on the unified-($q,s$) entanglement (UE). We provide the monogamy relation based on the squared UE for a reduced density matrix of a qudit GW state, as well as tighter monogamy relations based on the $\alpha$th ($\alpha \ge 2$) power of UE. Furthermore, for an $n$-qudit system $AB{C}_1\mathrm{.}\mathrm{.}\mathrm{.}{C}_{n-2}$, generalized monogamy relation and upper bound satisfied by the $\beta$th ($0\le \beta \le 1$) power of UE for the GW states under the partition $AB$ and $C_1\mathrm{.}\mathrm{.}\mathrm{.}{C}_{n-2}$ are established. In particular, two partition-dependent residual entanglements for the GW states are analyzed in detail.