Limit of the maximum random permutation set entropy

The Random Permutation Set (RPS) is a recently proposed new type of set, 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 to understand 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 proven. 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 the cardinality of a RPS (marked as N) approaches to infinity, the limit form of the envelope of the entropy of RPS converges to $e\cdot {\left(\text{N!}\right)}^2$, which is highly connected to the constant $e$ and factorial. Finally, numerical examples validate the efficiency and conciseness of the proposed envelope, which provides new insights into the maximum entropy function.