We survey known results and open problems in abelian combinatorics on words. Abelian
combinatorics on words is the extension to the commutative setting of the classical theory of …

Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2+
2/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent …

A word of length n is rich if it contains n nonempty palindromic factors. An infinite word is rich
if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word …

Combinatorics on words aims at finding deep connections between properties of
sequences. The resulting theoretical findings are often used in the design of efficient …

Regular episturmian words are episturmian words whose directive words have a regular
and restricted form making them behave more like Sturmian words than general episturmian …

A non-empty word w is a border of a word u if| w|<| u| and w is both a prefix and a suffix of u.
A word u is privileged if| u|≤ 1 or if u has a privileged border w that appears exactly twice in …

We study the abelian period sets of Sturmian words, which are codings of irrational rotations
on a one-dimensional torus. The main result states that the minimum abelian period of a …

Bell and Shallit recently introduced the Lie complexity of an infinite word s as the function
counting for each length the number of conjugacy classes of words whose elements are all …

We study the set of $ k $-abelian critical exponents of all Sturmian words. It has been proven
that in the case $ k= 1$ this set coincides with the Lagrange spectrum. Thus the sets …

The abelian critical exponent of an infinite word w is defined as the maximum ratio between
the exponent and the period of an abelian power occurring in w. It was shown by Fici et al …