TY - JOUR

T1 - Abelian-square-rich words

AU - Fici, Gabriele

AU - Mignosi, Filippo

AU - Shallit, Jeffrey

PY - 2017

Y1 - 2017

N2 - An abelian square is the concatenation of two words that are anagrams of one another. A word of length n can contain at most Θ(n2) distinct factors, and there exist words of length n containing Θ(n2) distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length n grows quadratically with n. More precisely, we say that an infinite word w is abelian-square-rich if, for every n, every factor of w of length n contains, on average, a number of distinct abelian-square factors that is quadratic in n; and uniformly abelian-square-rich if every factor of w contains a number of distinct abelian-square factors that is proportional to the square of its length. Of course, if a word is uniformly abelian-square-rich, then it is abelian-square-rich, but we show that the converse is not true in general. We prove that the Thue–Morse word is uniformly abelian-square-rich and that the function counting the number of distinct abelian-square factors of length 2n of the Thue–Morse word is 2-regular. As for Sturmian words, we prove that a Sturmian word sα of angle α is uniformly abelian-square-rich if and only if the irrational α has bounded partial quotients, that is, if and only if sα has bounded exponent.

AB - An abelian square is the concatenation of two words that are anagrams of one another. A word of length n can contain at most Θ(n2) distinct factors, and there exist words of length n containing Θ(n2) distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length n grows quadratically with n. More precisely, we say that an infinite word w is abelian-square-rich if, for every n, every factor of w of length n contains, on average, a number of distinct abelian-square factors that is quadratic in n; and uniformly abelian-square-rich if every factor of w contains a number of distinct abelian-square factors that is proportional to the square of its length. Of course, if a word is uniformly abelian-square-rich, then it is abelian-square-rich, but we show that the converse is not true in general. We prove that the Thue–Morse word is uniformly abelian-square-rich and that the function counting the number of distinct abelian-square factors of length 2n of the Thue–Morse word is 2-regular. As for Sturmian words, we prove that a Sturmian word sα of angle α is uniformly abelian-square-rich if and only if the irrational α has bounded partial quotients, that is, if and only if sα has bounded exponent.

KW - Abelian square

KW - Computer Science (all)

KW - Sturmian word

KW - Theoretical Computer Science

KW - Thue–Morse word

KW - Abelian square

KW - Computer Science (all)

KW - Sturmian word

KW - Theoretical Computer Science

KW - Thue–Morse word

UR - http://hdl.handle.net/10447/238201

UR - http://www.journals.elsevier.com/theoretical-computer-science/

M3 - Article

VL - 684

SP - 29

EP - 42

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -