Factorial Permutation Generation

Permutations are widely used in practice for solving problems of combinatorial optimization, noise-resistant coding, as well as in modern methods of protecting information as their structural elements. The paper poses the problem of accelerating the search of permutations with a changing interval between them in increasing or decreasing order. To solve this problem, we propose a method of permutations searching with a change in its direction and interval between permutations, using factorial numbers. The fundamentals of the factorial numbers theory are presented and a method for obtaining them is given. Examples of addition operations between factorial numbers are given. An improved method for converting factorial numbers into permutations is considered. It is shown that the proposed method of permutations searching has increased speed due to the use of arithmetic operations of addition and subtraction of factorial numbers, and also allows to change the direction of permutations searching, which increases its flexibility. Due to its simplicity and regularity of the structure, the proposed method is effective for both software and hardware implementations. It is promising for solving problems both with a complete enumeration of all their solutions, and with partial ones.