Enumeration of symmetrical necklaces

The problem of the symmetrical stanzas enumeration, whose models are necklaces or simple cycles of a given length, is considered. The automorphisms of the necklace symmetry group form a dihedral group, whose action on the set of necklaces allows us to distribute them over orbits. With help of a specially developed algorithm, the orbits are divided into sets of symmetric and asymmetric necklaces, while the symmetrical necklaces are classified into types and classes of symmetry. Mirror or axial symmetry, as well as its variants, achiral and chiral symmetries, are considered. Analytical results are given for calculating the cardinality of symmetry classes (of representative graphs) belonging to the same type of symmetry. Applications of the resulting classification to the analysis of octets, for which various rhyming schemes are used as a attribute of symmetry, are considered. The analysis of the applicability of symmetrical stanzas in the practice of versification has been carried out. It is shown that within the framework of the proposed approach, all known hard forms of versification for octets, such as the triolet, siciliane, octave, and others, are symmetrical; examples of rarely used symmetrical stanzas are listed, indicating the possible reasons for this phenomenon. The possibility of using the proposed approach in the problem of choosing the rhythmic profile of a stanza, which generates the symmetry of the rhythm, is shown. The author declares no conflicts of interests. The author declares no conflicts of interests.

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