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In the area of pattern avoidability the central role is played by special words called Zimin patterns. The symbols of these patterns are treated as variables and the rank of the pattern is its number of variables. Zimin type of a word $x$ is introduced here as the maximum rank of a Zimin pattern matching $x$. We show how to compute Zimin type of a word on-line in linear time. Consequently we get a quadratic time, linear-space algorithm for searching Zimin patterns in words. Then we how the Zimin type of the length $n$ prefix of the infinite Fibonacci word is related to the representation of $n$ in the Fibonacci numeration system. Using this relation, we prove that Zimin types of such prefixes and Zimin patterns inside them can be found in logarithmic time. Finally, we give some bounds on the function $f(n,k)$ such that every $k$-ary word of length at least $f(n,k)$ has a factor that matches the rank $n$ Zimin pattern.
A generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as a nonincr
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We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.