James D. Currie

Pattern avoidance: themes and variations

We review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems.

Open problems in pattern avoidance

In this department the MONTHLYpresents easily stated unsolved problems dealing with notions ordinarily encountered in undergraduate mathematics. Each problem should be accompanied by relevant references (if any are known to the author) and by a brief description of known partial or related results. Typescripts should be sent to Richard Guy, Department of Mathematics & Statistics, The University of Calgary, Alberta, Canada T2N 1N4.

A proof of Dejean’s conjecture

We prove Dejeans conjecture. Specifically, we show that Dejeans conjecture holds for the last remaining open values of n, namely 15 < n < 26.

Dejean's conjecture and Sturmian words

Dejean conjectured that the repetition threshold of a k-letter alphabet is k/(k-1),k 3,4. Values of the repetition threshold for k<5 were found by Thue, Dejean and Pansiot. Moulin-Ollagnier attacked Dejeans conjecture for 5@?k@?11. Building on the work of Moulin-Ollagnier, we propose a method for deciding whether a given Sturmian word with quadratic slope confirms the conjecture for a given k. Elaborating this method in terms of directive words, we develop a search algorithm for verifying the conjecture for a given k. An implementation of our algorithm gives suitable Sturmian words for 7@?k@?14. We prove that for k=5, no suitable Sturmian word exists.

LEAST PERIODS OF FACTORS OF INFINITE WORDS

Work of the first author supported by a Discovery Grant from NSERC. Work of the second author supported by the Finnish Academy under grant 8206039.

Dejean's conjecture holds for n≥30

Abstract We extend Carpi’s results by showing that Dejean’s conjecture holds for n ≥ 30 .

Avoiding Three Consecutive Blocks of the Same Size and Same Sum

We show that there exists an infinite word over the alphabet {0, 1, 3, 4} containing no three consecutive blocks of the same size and the same sum. This answers an open problem of Pirillo and Varricchio from 1994.

DEJEAN'S CONJECTURE HOLDS FOR N > 27

We show that Dejean’s conjecture holds for n ≥ 27. This brings the final resolution of the conjecture by the approach of Moulin Ollagnier within range of the computationally feasible.

A cyclic binary morphism avoiding Abelian fourth powers

We exhibit a cyclic binary morphism avoiding Abelian fourth powers.

Unary patterns with involution

An infinite word w avoids a pattern p with the involution if there is no substitution for the variables in p and no involution on substituted variables such that the resulting word is a factor of w. An avoidance index of pattern p is the smallest alphabet size for which a word exists such that p is avoided. A pattern is called unary, if only one variable occurs in it. In this paper, we give the avoidance indices for all unary patterns with involution.