173,472 research outputs found
Avoiding 3/2-powers over the natural numbers
In this paper we answer the following question: what is the lexicographically
least sequence over the natural numbers that avoids 3/2-powers?Comment: 9 page
Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions
Two well studied Ramsey-theoretic problems consider subsets of the natural
numbers which either contain no three elements in arithmetic progression, or in
geometric progression. We study generalizations of this problem, by varying the
kinds of progressions to be avoided and the metrics used to evaluate the
density of the resulting subsets. One can view a 3-term arithmetic progression
as a sequence , where , a nonzero
integer. Thus avoiding three-term arithmetic progressions is equivalent to
containing no three elements of the form with , the set of integer translations. One can similarly
construct related progressions using different families of functions. We
investigate several such families, including geometric progressions ( with a natural number) and exponential progressions ().
Progression-free sets are often constructed "greedily," including every
number so long as it is not in progression with any of the previous elements.
Rankin characterized the greedy geometric-progression-free set in terms of the
greedy arithmetic set. We characterize the greedy exponential set and prove
that it has asymptotic density 1, and then discuss how the optimality of the
greedy set depends on the family of functions used to define progressions.
Traditionally, the size of a progression-free set is measured using the (upper)
asymptotic density, however we consider several different notions of density,
including the uniform and exponential densities.Comment: Version 1.0, 13 page
Anti-Powers in Infinite Words
In combinatorics of words, a concatenation of consecutive equal blocks is
called a power of order . In this paper we take a different point of view
and define an anti-power of order as a concatenation of consecutive
pairwise distinct blocks of the same length. As a main result, we show that
every infinite word contains powers of any order or anti-powers of any order.
That is, the existence of powers or anti-powers is an unavoidable regularity.
Indeed, we prove a stronger result, which relates the density of anti-powers to
the existence of a factor that occurs with arbitrary exponent. As a
consequence, we show that in every aperiodic uniformly recurrent word,
anti-powers of every order begin at every position. We further show that every
infinite word avoiding anti-powers of order is ultimately periodic, while
there exist aperiodic words avoiding anti-powers of order . We also show
that there exist aperiodic recurrent words avoiding anti-powers of order .Comment: Revision submitted to Journal of Combinatorial Theory Series
On sets of integers which contain no three terms in geometric progression
The problem of looking for subsets of the natural numbers which contain no
3-term arithmetic progressions has a rich history. Roth's theorem famously
shows that any such subset cannot have positive upper density. In contrast,
Rankin in 1960 suggested looking at subsets without three-term geometric
progressions, and constructed such a subset with density about 0.719. More
recently, several authors have found upper bounds for the upper density of such
sets. We significantly improve upon these bounds, and demonstrate a method of
constructing sets with a greater upper density than Rankin's set. This
construction is optimal in the sense that our method gives a way of effectively
computing the greatest possible upper density of a geometric-progression-free
set. We also show that geometric progressions in Z/nZ behave more like Roth's
theorem in that one cannot take any fixed positive proportion of the integers
modulo a sufficiently large value of n while avoiding geometric progressions.Comment: 16 page
On partitions avoiding 3-crossings
A partition on has a crossing if there exists
such that and are in the same block, and are in the
same block, but and are not in the same block. Recently, Chen et
al. refined this classical notion by introducing -crossings, for any integer
. In this new terminology, a classical crossing is a 2-crossing. The number
of partitions of avoiding 2-crossings is well-known to be the th
Catalan number . This raises the question of
counting -noncrossing partitions for . We prove that the sequence
counting 3-noncrossing partitions is P-recursive, that is, satisfies a linear
recurrence relation with polynomial coefficients. We give explicitly such a
recursion. However, we conjecture that -noncrossing partitions are not
P-recursive, for
Identifying all abelian periods of a string in quadratic time and relevant problems
Abelian periodicity of strings has been studied extensively over the last
years. In 2006 Constantinescu and Ilie defined the abelian period of a string
and several algorithms for the computation of all abelian periods of a string
were given. In contrast to the classical period of a word, its abelian version
is more flexible, factors of the word are considered the same under any
internal permutation of their letters. We show two O(|y|^2) algorithms for the
computation of all abelian periods of a string y. The first one maps each
letter to a suitable number such that each factor of the string can be
identified by the unique sum of the numbers corresponding to its letters and
hence abelian periods can be identified easily. The other one maps each letter
to a prime number such that each factor of the string can be identified by the
unique product of the numbers corresponding to its letters and so abelian
periods can be identified easily. We also define weak abelian periods on
strings and give an O(|y|log(|y|)) algorithm for their computation, together
with some other algorithms for more basic problems.Comment: Accepted in the "International Journal of foundations of Computer
Science
Patterns in Inversion Sequences I
Permutations that avoid given patterns have been studied in great depth for
their connections to other fields of mathematics, computer science, and
biology. From a combinatorial perspective, permutation patterns have served as
a unifying interpretation that relates a vast array of combinatorial
structures. In this paper, we introduce the notion of patterns in inversion
sequences. A sequence is an inversion sequence if for all . Inversion sequences of length are in
bijection with permutations of length ; an inversion sequence can be
obtained from any permutation by setting . This correspondence makes it
a natural extension to study patterns in inversion sequences much in the same
way that patterns have been studied in permutations. This paper, the first of
two on patterns in inversion sequences, focuses on the enumeration of inversion
sequences that avoid words of length three. Our results connect patterns in
inversion sequences to a number of well-known numerical sequences including
Fibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers
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