Sesquipower

Formally, let A be an alphabet and A^∗ be the free monoid of finite strings over A. Every non-empty word w in A⁺ is a sesquipower of order 1. If u is a sesquipower of order n then any word w = uvu is a sesquipower of order n + 1.[1] The degree of a non-empty word w is the largest integer d such that w is a sesquipower of order d.[2]

A bi-ideal sequence is a sequence of words f_i where f₁ is in A⁺ and

${\displaystyle f_{i+1}=f_{i}g_{i}f_{i}\ }$

for some g_i in A^∗ and i ≥ 1. The degree of a word w is thus the length of the longest bi-ideal sequence ending in w.[2]

For a finite alphabet A on k letters, there is an integer M depending on k and n, such that any word of length M has a factor which is a sesquipower of order at least n. We express this by saying that the sesquipowers are unavoidable patterns.[3][4]

Given an infinite bi-ideal sequence, we note that each f_i is a prefix of f_i+1 and so the f_i converge to an infinite sequence

${\displaystyle f=f_{1}g_{1}f_{1}g_{2}f_{1}g_{1}f_{1}g_{3}f_{1}\cdots \ }$

We define an infinite word to be a sesquipower if it is the limit of an infinite bi-ideal sequence.[5] An infinite word is a sesquipower if and only if it is a recurrent word,[5][6] that is, every factor occurs infinitely often.[7]

Fix a finite alphabet A and assume a total order on the letters. For given integers p and n, every sufficiently long word in A^∗ has either a factor which is a p-power or a factor which is an n-sesquipower; in the latter case the factor has an n-factorisation into Lyndon words.[6]

• ABACABA pattern

• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043.
• Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.