384 research outputs found
On Long Words Avoiding Zimin Patterns
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern p is unavoidable if, over every finite alphabet, every sufficiently long word encounters p. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over n distinct variables is unavoidable if, and only if, p itself is encountered in the n-th Zimin pattern. Given an alphabet size k, we study the minimal length f(n,k) such that every word of length f(n,k) encounters the n-th Zimin pattern. It is known that f is upper-bounded by a tower of exponentials. Our main result states that f(n,k) is lower-bounded by a tower of n-3 exponentials, even for k=2. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense
Tower-type bounds for unavoidable patterns in words
A word is said to contain the pattern if there is a way to substitute
a nonempty word for each letter in so that the resulting word is a subword
of . Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised
the patterns which are unavoidable, in the sense that any sufficiently long
word over a fixed alphabet contains . Zimin's characterisation says that a
pattern is unavoidable if and only if it is contained in a Zimin word, where
the Zimin words are defined by and . We
study the quantitative aspects of this theorem, obtaining essentially tight
tower-type bounds for the function , the least integer such that any
word of length over an alphabet of size contains . When , the first non-trivial case, we determine up to a constant factor,
showing that .Comment: 17 page
Avoidability of formulas with two variables
In combinatorics on words, a word over an alphabet is said to
avoid a pattern over an alphabet of variables if there is no
factor of such that where is a
non-erasing morphism. A pattern is said to be -avoidable if there exists
an infinite word over a -letter alphabet that avoids . We consider the
patterns such that at most two variables appear at least twice, or
equivalently, the formulas with at most two variables. For each such formula,
we determine whether it is -avoidable, and if it is -avoidable, we
determine whether it is avoided by exponentially many binary words
Toward the Combinatorial Limit Theory of Free Words
Free words are elements of a free monoid, generated over an alphabet via the
binary operation of concatenation. Casually speaking, a free word is a finite
string of letters. Henceforth, we simply refer to them as words. Motivated by
recent advances in the combinatorial limit theory of graphs-notably those
involving flag algebras, graph homomorphisms, and graphons-we investigate the
extremal and asymptotic theory of pattern containment and avoidance in words.
Word V is a factor of word W provided V occurs as consecutive letters within W.
W is an instance of V provided there exists a nonerasing monoid homomorphsism
{\phi} with {\phi}(V) = W. For example, using the homomorphism {\phi} defined
by {\phi}(P) = Ror, {\phi}(h) = a, and {\phi}(D) = baugh, we see that Rorabaugh
is an instance of PhD.
W avoids V if no factor of W is an instance of V. V is unavoidable provided,
over any finite alphabet, there are only finitely many words that avoid V.
Unavoidable words were classified by Bean, Ehrenfeucht, and McNulty (1979) and
Zimin (1982). We briefly address the following Ramsey-theoretic question: For
unavoidable word V and a fixed alphabet, what is the longest a word can be that
avoids V? The density of V in W is the proportion of nonempty substrings of W
that are instances of V. Since there are 45 substrings in Rorabaugh and 28 of
them are instances of PhD, the density of PhD in Rorabaugh is 28/45. We
establish a number of asymptotic results for word densities, including the
expected density of a word in arbitrarily long, random words and the minimum
density of an unavoidable word over arbitrarily long words.
This is joint work with Joshua Cooper.Comment: 110 pages, dissertatio
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
Crucial and bicrucial permutations with respect to arithmetic monotone patterns
A pattern is a permutation, and an arithmetic occurrence of in
(another) permutation is a subsequence
of that is order isomorphic to
where the numbers form an arithmetic progression. A
permutation is -crucial if it avoids arithmetically the patterns
and but its extension to the right by any element
does not avoid arithmetically these patterns. A -crucial permutation
that cannot be extended to the left without creating an arithmetic occurrence
of or is called -bicrucial.
In this paper we prove that arbitrary long -crucial and
-bicrucial permutations exist for any . Moreover, we
show that the minimal length of a -crucial permutation is
, while the minimal length of a
-bicrucial permutation is at most ,
again for
- …