I. Grebennik, O.Chorna

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Abstract. The objective of this paper is the investigation of the cyclic structure and permutation properties based on neighbor elements transposition properties and the properties of the permutation polyhedron. In this paper we consider special type of transpositions of elements in a permutation. A feature of these transpositions is that they corresponding to the adjacency criterion in a permutation polyhedron. We will investigate permutation properties with the help of the permutation polyhedron by using the immersing in the Euclidian space. Six permutation types are considered in correspondence with the location of arbitrary components. We consider the impact of the corresponding components on the cyclic structure of permutations depending on the type of a permutation. In this paper we formulate the assertion about the features of the impact of transpositions corresponding to the adjacency criterion on the permutations consisting of the one cycle. During the proof of statement all six types of permutations are considered and clearly demonstrated that only two types arrangement of the elements in the cycle contribute to the persistence a single cycle in the permutation after the impact of two transpositions. Research conducted in the given work, will be further employed in mathematical modeling and computational methods. Especially for solving combinatorial optimization problems and for the generation of combinatorial objects with a predetermined cyclic structure.

Key words. Permutations set, permutation polyhedron, adjacency criterion, permutation properties, transposition, combinatorics.

ELEMENTS TRANSPOSITIONS AND THEIR IMPACT ON THE CYCLIC STRUCTURE OF PERMUTATIONS

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