Sesquipower

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

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

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]

Unavoidable patterns

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

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]

See also

• ABACABA pattern