A d-step approach for distinct squares in strings

Abstract

We present an approach to the problem of maximum number of distinct squares in a string which underlines the importance of considering as key variables both the length n and n - d where d is the size of the alphabet. We conjecture that a string of length n and containing d distinct symbols has no more than n - d distinct squares, show the critical role played by strings satisfying n = 2d, and present some properties satisfied by strings of length bounded by a constant times the size of the alphabet. © 2011 Springer-Verlag.