Abelian Primitive Words

We investigate Abelian primitive words, which are words that are not Abelian powers. We show that unlike classical primitive words, the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive in linear time. Also different from classical primitive words, we find that a word may have more than one Abelian root. We also consider enumeration problems and the relation to the theory of codes

Infinite permutations vs. infinite words

I am going to compare well-known properties of infinite words with those of infinite permutations, a new object studied since middle 2000s. Basically, it was Sergey Avgustinovich who invented this notion, although in an early study by Davis et al. permutations appear in a very similar framework as early as in 1977. I am going to tell about periodicity of permutations, their complexity according to several definitions and their automatic properties, that is, about usual parameters of words, now extended to permutations and behaving sometimes similarly to those for words, sometimes not. Another series of results concerns permutations generated by infinite words and their properties. Although this direction of research is young, many people, including two other speakers of this meeting, have participated in it, and I believe that several more topics for further study are really promising.Comment: In Proceedings WORDS 2011, arXiv:1108.341

A Generalization of the Genocchi Numbers with Applications to Enumeration of Finite Automata

We consider a natural generalization of the well-studied Genocchi numbers. This generalization proves useful in enumerating the class of deterministic finite automata (DFA) which accept a finite language. We also link our generalization to the method of Gandhi polynomials for generating Genocchi numbers

Deletion along Trajectories

We describe a new way to model deletions on formal languages, called deletion along trajectories. We examine its closure properties, and show that it serves as an inverse to shuffle on trajectories, recently introduced by Mateescu et al. This leads to results on the decidability of equations of the form L T X = R, where L; R are regular languages and X is unknown.