198 research outputs found
On Generating Binary Words Palindromically
We regard a finite word up to word isomorphism as an
equivalence relation on where is equivalent to if
and only if Some finite words (in particular all binary words) are
generated by "{\it palindromic}" relations of the form for some
choice of and That is to say,
some finite words are uniquely determined up to word isomorphism by the
position and length of some of its palindromic factors. In this paper we study
the function defined as the least number of palindromic relations
required to generate We show that every aperiodic infinite word must
contain a factor with and that some infinite words have
the property that for each factor of We obtain a
complete classification of such words on a binary alphabet (which includes the
well known class of Sturmian words). In contrast for the Thue-Morse word, we
show that the function is unbounded
Abelian bordered factors and periodicity
A finite word u is said to be bordered if u has a proper prefix which is also
a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that
an infinite word is purely periodic if and only if it contains only finitely
many unbordered factors. We are interested in abelian and weak abelian
analogues of this result; namely, we investigate the following question(s): Let
w be an infinite word such that all sufficiently long factors are (weakly)
abelian bordered; is w (weakly) abelian periodic? In the process we answer a
question of Avgustinovich et al. concerning the abelian critical factorization
theorem.Comment: 14 page
- …