Picture of athlete cycling

Open Access research with a real impact on health...

The Strathprints institutional repository is a digital archive of University of Strathclyde's Open Access research outputs. Strathprints provides access to thousands of Open Access research papers by Strathclyde researchers, including by researchers from the Physical Activity for Health Group based within the School of Psychological Sciences & Health. Research here seeks to better understand how and why physical activity improves health, gain a better understanding of the amount, intensity, and type of physical activity needed for health benefits, and evaluate the effect of interventions to promote physical activity.

Explore open research content by Physical Activity for Health...

On shortest crucial words avoiding abelian powers

Avgustinovich, Sergey and Glen, Amy and Halldorsson, Bjarni and Kitaev, Sergey (2010) On shortest crucial words avoiding abelian powers. Discrete Applied Mathematics, 158 (6). pp. 605-607. ISSN 0166-218X

Full text not available in this repository. Request a copy from the Strathclyde author

Abstract

Let k≥2k≥2 be an integer. An abelian kkth power is a word of the form X1X2⋯XkX1X2⋯Xk where XiXi is a permutation of X1X1 for 2≤i≤k2≤i≤k. A word WW is said to be crucial with respect to abelian kkth powers if WW avoids abelian kkth powers, but WxWx ends with an abelian kkth power for any letter xx occurring in WW. Evdokimov and Kitaev (2004) [2] have shown that the shortest length of a crucial word on nn letters avoiding abelian squares is 4n−74n−7 for n≥3n≥3. Furthermore, Glen et al. (2009) [3] proved that this length for abelian cubes is 9n−139n−13 for n≥5n≥5. They have also conjectured that for any k≥4k≥4 and sufficiently large nn, the shortest length of a crucial word on nn letters avoiding abelian kkth powers, denoted by ℓk(n)ℓk(n), is k2n−(k2+k+1)k2n−(k2+k+1). This is currently the best known upper bound for ℓk(n)ℓk(n), and the best known lower bound, provided in Glen et al., is 3kn−(4k+1)3kn−(4k+1) for n≥5n≥5 and k≥4k≥4. In this note, we improve this lower bound by proving that for n≥2k−1n≥2k−1, ℓk(n)≥k2n−(2k3−3k2+k+1)ℓk(n)≥k2n−(2k3−3k2+k+1); thus showing that the aforementioned conjecture is true asymptotically (up to a constant term) for growing nn.