Алгоритмы перечисления симметричных хордовых диаграмм

Abstract

Numerous applications require exhaustive lists of strings, which may be subject to various requirements, for example, such as their non-equivalence under the action of symmetry groups on them. The necklace is the equivalence class of r-ary strings under rotation. An unlabeled r-ary necklace is an equivalence class of r-ary strings under rotation and permutation of alphabetic characters. Let G be a symmetry group acting on a given set of necklaces T. A necklace x  T is called symmetric if there exists an element g  G, g  e such that gx = x. Other definitions of the symmetry of necklaces are possible, equivalent to the above. All of them, in one way or another, are involved in determining the symmetry of the figure compared to the unlabelled necklace. A flat figure is called symmetrical if it is self-aligned during movements of space R2 i. e. during its isometric transformations. A special case of figures and, accordingly, necklaces are chord diagrams. Chord diagrams represent an object of research that is interesting from different aspects (knot theory, Feynman diagrams, representations of Lie algebras), and has been studied by many authors. The article presents algorithms for enumerating symmetric chord diagrams and shows their connection with the knot theory using simple examples. The list of symmetrical chord diagrams may, in particular, be of interest in the field of mathematical poetry, which studies the dynamics of poetic stanzas horizontally (rhythm, syllabic volume of verses) and vertically (rhyme schemes, refrains, and other stylistic devices).