548 research outputs found
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
On a generalization of Abelian equivalence and complexity of infinite words
In this paper we introduce and study a family of complexity functions of
infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+
\cup {+\infty} and be a finite non-empty set. Two finite words and
in are said to be -Abelian equivalent if for all of length
less than or equal to the number of occurrences of in is equal to
the number of occurrences of in This defines a family of equivalence
relations on bridging the gap between the usual notion of
Abelian equivalence (when ) and equality (when We show that
the number of -Abelian equivalence classes of words of length grows
polynomially, although the degree is exponential in Given an infinite word
\omega \in A^\nats, we consider the associated complexity function \mathcal
{P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of
-Abelian equivalence classes of factors of of length We show
that the complexity function is intimately linked with
periodicity. More precisely we define an auxiliary function q^k: \nats
\rightarrow \nats and show that if for
some k \in \ints ^+ \cup {+\infty} and the is ultimately
periodic. Moreover if is aperiodic, then if and only if is Sturmian. We also
study -Abelian complexity in connection with repetitions in words. Using
Szemer\'edi's theorem, we show that if has bounded -Abelian
complexity, then for every D\subset \nats with positive upper density and for
every positive integer there exists a -Abelian power occurring in
at some position $j\in D.
When is a Schubert variety Gorenstein?
A (normal) variety is Gorenstein if it is Cohen-Macualay and its canonical
sheaf is a line bundle. This property, which measures the ``pathology'' of the
singularities of a variety, is thus stronger than Cohen-Macualayness, but is
also weaker than smoothness. We determine which Schubert varieties are
Gorenstein in terms of a combinatorial characterization using generalized
pattern avoidance conditions. We also give an explicit description as a line
bundle of the canonical sheaf of a Gorenstein Schubert variety.Comment: 15 pages, geometric characterization of Gorensteinness added; final
version to appear in Adv. Mat
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced
Impartial achievement and avoidance games for generating finite groups
We study two impartial games introduced by Anderson and Harary and further
developed by Barnes. Both games are played by two players who alternately
select previously unselected elements of a finite group. The first player who
builds a generating set from the jointly selected elements wins the first game.
The first player who cannot select an element without building a generating set
loses the second game. After the development of some general results, we
determine the nim-numbers of these games for abelian and dihedral groups. We
also present some conjectures based on computer calculations. Our main
computational and theoretical tool is the structure diagram of a game, which is
a type of identification digraph of the game digraph that is compatible with
the nim-numbers of the positions. Structure diagrams also provide simple yet
intuitive visualizations of these games that capture the complexity of the
positions.Comment: 28 pages, 44 figures. Revised in response to comments from refere
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
Pattern avoidance: themes and variations
AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems
On the Parikh-de-Bruijn grid
We introduce the Parikh-de-Bruijn grid, a graph whose vertices are
fixed-order Parikh vectors, and whose edges are given by a simple shift
operation. This graph gives structural insight into the nature of sets of
Parikh vectors as well as that of the Parikh set of a given string. We show its
utility by proving some results on Parikh-de-Bruijn strings, the abelian analog
of de-Bruijn sequences.Comment: 18 pages, 3 figures, 1 tabl
Avoiding Patterns in the Abelian Sense
We classify all 3 letter patterns that are avoidable in the abelian sense. A short list of four letter patterns for which abelian avoidance is undecided is given. Using a generalization of Zimin words we deduce some properties of ω-words avoiding these patterns.Research of both authors supported by NSERC Operating Grants.https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/avoiding-patterns-in-the-abelian-sense/42148B0781A38A6618A537AAD7D39B8
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