Classification Of Permutations Research Articles

On Classification of Permutations

On Classification of Permutations

On affine classification of permutations on the space GF(2)3

Abstract We give an elementary proof that by multiplication on left and right by affine permutations A, B ∈ AGL(3, 2) each permutation π : GF(2)3 → GF(2)3 may be reduced to one of the 4 permutations for which the 3 × 3-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory.

Кодування і класифікація перестановок за спеціальним перетворенням з оцінками потужностей класів

Scientific articles investigating properties and estimates of the number of so-called complete permutations are surveyed and analyzed. The paper introduces a special S-transform on the set of permutations and determines the permutation properties according to this transform. Classification and coding of permutations by equivalence classes according to their properties with respect to S-transformation is proposed. This classification and permutation properties, in particular, generalize known results for complete permutations regarding determining certain cryptographic properties of substitutions that affect the cryptographic transformations security. The exact values of the number of permutations in equivalence classes for certain permutation sizes are calculated and the estimates of the cardinality of classes with various properties are constructed by statistical modeling. The complete list of permutation classes with the exact values of their sizes for permutations of order n = 11 is presented. The interval estimates for the size of classes with various characteristics for permutations of order n = 11, 26, 30, 31, 32, 33, 45, 55 are obtained. Monte Carlo estimates and bounds of confidence intervals used the approximation of the binomial distribution by the normal and Poisson distributions, as well as the Python programming language package Scipy. Statistical tables have been calculated that can be used for further conclusions and estimates. The classification of permutations by their properties with respect to the introduced transform can be used in constructing high-quality cryptographic transformations and transformations with special features. The classes of complete permutations with their properties are selected as the best for rotary cryptosystems applications. The obtained results can be used, in particular, to search for permutations with certain characteristics and properties, to find the probability that the characteristic of the generated permutation belongs to a collection of given characteristics, to estimate the complexity of finding permutations with certain properties. A statistical criterion of consent, which uses the characteristics of permutations by S-transformation to test the generators of random permutations and substitutions is proposed.

Bernoulli, Euler, permutations and quantum algebras

The combinatorial properties of the Bernoulli and Euler numbers are interpreted using a new classification of permutations. The classification is naturally described by an operator algebra of a type familiar from quantum theory. It has a duality structure described by an operator satisfying anticommutation relations.

Classification of permutations and cycles of maximum topological entropy

Iff is a continuous self-map of a compact interval we can represent each finite fully invariant set off by a permutation. We can then calculate the topological entropy of the permutation. This provides us with a numerical measure of complexity for any map which exhibits a given permutation type. In this paper we present cyclic and noncyclic permutations which have maximum topological entropy amongst all cyclic or noncyclic permutations of the same length.

On the transformation and classification of Permutations

Let represent any permutation of the first n natural numbers. If we take the first number, a1, and put it last, we get a new permutation; and if we perform this operation (n−1) times, we get the (n−1) permutations