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In mathematics, a sesquipower or Zimin word is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one.

Formal definition[edit]

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]

Bi-ideal sequence[edit]

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

for some gi 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]

Unavoidable patterns[edit]

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]

Sesquipowers in infinite sequences[edit]

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

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]

See also[edit]


  1. ^ Lothaire (2011) p. 135
  2. ^ a b Lothaire (2011) p. 136
  3. ^ Lothaire (2011) p. 137
  4. ^ Berstel et al (2009) p.132
  5. ^ a b Lothiare (2011) p. 141
  6. ^ a b Berstel et al (2009) p.133
  7. ^ Lothaire (2011) p. 30
  • Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. Vol. 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. Vol. 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. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.