On a Special Class of Primitive Words

Abstract

Abstract. When representing DNA molecules as words, it is necessary to take into account the fact that a word u encodes basically the same information as its Watson-Crick complement θ(u), where θ denotes the Watson-Crick complementarity function. Thus, an expression which involves only a word u and its complement can be still considered as a repeating sequence. In this context, we define and investigate the properties of a special class of primitive words, called θ-primitive, which cannot be expressed as such repeating sequences. For instance, we prove the existence of a unique θ-primitive root of a given word, and we give some constraints forcing two distinct words to share their θ-primitive root. Also, we present an extension of the well-known Fine and Wilf Theorem, for which we give an optimal bound.

Equations enforcing repetitions under permutations

“…In the case of antimorphic, we show, in Section 4, that the equation ( ) ( ) = ( ) , parametrised by the non-empty words and and the integers , , , may have as solutions words which are not [ ]-repetitions. Thus, following the results of [8,7], we characterise exactly the paramterized equations (or, in other words, find all the parameters/exponents 1 , 2 , 3 , 1 , and 2 )…”

“…Theorem 1 was extended in [10] for [ ]-repetitions. Theorem 2 was extended in the setting of anti-/morphic involutions in [7]. Theorem 3 was extended for [ ]-repetitions where is an antimorphic involution in a series of papers that culminated in [9], where a full characterisation of the triples ( , , ) for which the equation…”

“…Pseudo-repetitions were introduced [6,7], as a generalisation of classical repetitions, inspired by molecular biology. A word is a pseudo-repetition, or more precisely an -repetition, if it equals a repeated concatenation of one of its prefixes and its image ( ) under some morphism or antimorphism (for short "anti-/morphism") , i.e.…”

“…is in { , ( )} + . To fit the biological motivation, in [7] was defined as an antimorphic involution, i.e. 2 ( ) = for all words .…”

“…More interesting to us, if is not restricted, pseudo-repetitions generalise not only repetitions (when is the identity morphism), but also palindromes (when is the mirror image); both these concepts are central in combinatorics on words, so their generalisations are of intrinsic theoretical interest. Initial results (see [7] and [8]) concerned generalisations of the Theorem of Fine and Wilf, of the Lyndon and Schützenberger equation, and of other repetition enforcing results to the setting of -repetitions for antimorphic involutions . For instance, Czeizler et al [8] introduced a generalisation of Lyndon and Schützenberger's equations of a different kind.…”