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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 sequipower 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 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