Avoiding Squares and Overlaps Over the Natural Numbers
Abstract
We consider avoiding squares and overlaps over the natural numbers, using a greedy algorithm that chooses the least possible integer at each step; the word generated is lexicographically least among all such infinite words. In the case of avoiding squares, the word is 01020103..., the familiar ruler function, and is generated by iterating a uniform morphism. The case of overlaps is more challenging. We give an explicitlydefined morphism phi : N* > N* that generates the lexicographically least infinite overlapfree word by iteration. Furthermore, we show that for all h,k in N with h <= k, the word phi^{kh}(h) is the lexicographically least overlapfree word starting with the letter h and ending with the letter k, and give some of its symmetry properties.
 Publication:

arXiv eprints
 Pub Date:
 January 2009
 arXiv:
 arXiv:0901.1397
 Bibcode:
 2009arXiv0901.1397G
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Formal Languages and Automata Theory;
 68R15
 EPrint:
 16 pages, 2 tables