The Random Permutation Set (RPS) is a new type of set proposed recently, which can be regarded as the generalization of evidence theory. To measure the uncertainty of RPS, the entropy of RPS and its corresponding maximum entropy have been proposed. Exploring the maximum entropy provides a possible way of understanding the physical meaning of RPS. In this paper, a new concept, the envelope of entropy function, is defined. In addition, the limit of the envelope of RPS entropy is derived and proved. Compared with the existing method, the computational complexity of the proposed method to calculate the envelope of RPS entropy decreases greatly. The result shows that when $N \to \infty$, the limit form of the envelope of the entropy of RPS converges to $e \times (N!)^2$, which is highly connected to the constant $e$ and factorial. Finally, numerical examples validate the efficiency and conciseness of the proposed envelope, which provides a new insight into the maximum entropy function.

翻译：随机排列集（RPS）是近期提出的一种新型集合，可被视为证据理论的推广。为度量RPS的不确定性，已有研究定义了RPS熵及其对应的最大熵。探索最大熵为理解RPS的物理意义提供了可能途径。本文定义了一个新概念——熵函数包络线，并推导证明了RPS熵包络线的极限形式。与现有方法相比，所提方法计算RPS熵包络线的计算复杂度大幅降低。结果表明，当$N \to \infty$时，RPS熵包络线的极限形式收敛于$e \times (N!)^2$，该结果与常数$e$和阶乘高度相关。最后，数值实例验证了所提包络线的有效性与简洁性，为最大熵函数提供了新见解。