Search results for "FIBONACCI"

A NEW COMPLEXITY FUNCTION FOR WORDS BASED ON PERIODICITY

2013

Motivated by the extension of the critical factorization theorem to infinite words, we study the (local) periodicity function, i.e. the function that, for any position in a word, gives the size of the shortest square centered in that position. We prove that this function characterizes any binary word up to exchange of letters. We then introduce a new complexity function for words (the periodicity complexity) that, for any position in the word, gives the average value of the periodicity function up to that position. The new complexity function is independent from the other commonly used complexity measures as, for instance, the factor complexity. Indeed, whereas any infinite word with bound…

Complex Numbers and Polynomials

2016

As mentioned in Chap. 1, for a given set and an operator applied to its elements, if the result of the operation is still an element of the set regardless of the input of the operator, then the set is said closed with respect to that operator.

Permutation properties and the fibonacci semigroup

1989

A Loopless Generation of Bitstrings without p Consecutive Ones

2001

Let F n (p) be the set of all n-length bitstrings such that there are no p consecutive ls. F n (p) is counted with the pth order Fibonacci numbers and it may be regarded as the subsets of {1, 2,…, n} without p consecutive elements and bitstrings in F n (p) code a particular class of trees or compositions of an integer. In this paper we give a Gray code for F n (p) which can be implemented in a recursive generating algorithm, and finally in a loopless generating algorithm.

Characteristic Sturmian words are extremal for the Critical Factorization Theorem

2012

We prove that characteristic Sturmian words are extremal for the Critical Factorization Theorem (CFT) in the following sense. If p x ( n ) denotes the local period of an infinite word x at point n , we prove that x is a characteristic Sturmian word if and only if p x ( n ) is smaller than or equal to n + 1 for all n ≥ 1 and it is equal to n + 1 for infinitely many integers n . This result is extremal with respect to the \{CFT\} since a consequence of the \{CFT\} is that, for any infinite recurrent word x, either the function p x is bounded, and in such a case x is periodic, or p x ( n ) ≥ n + 1 for infinitely many integers n . As a byproduct of the techniques used in the paper we extend a r…

Diffraction by m-bonacci gratings

2015

We present a simple diffraction experiment with m-bonacci gratings as a new interesting generalization of the Fibonacci ones. Diffraction by these nonconventional structures is proposed as a motivational strategy to introduce students to basic research activities. The Fraunhofer diffraction patterns are obtained with the standard equipment present in most undergraduate physics labs and are compared with those obtained with regular periodic gratings. We show that m-bonacci gratings produce discrete Fraunhofer patterns characterized by a set of diffraction peaks which positions are related to the concept of a generalized golden mean. A very good agreement is obtained between experimental and …

Combinatorial isomorphism between Fibonacci classes

2008

Abstract In 1985 Simion and Schmidt showed that the set S n (T 3) of length n permutations avoiding the set of patterns T 3={123, 132, 213} is counted by (the second order) Fibonacci numbers. They also presented a constructive bijection between the set F n–1 of length (n–1) binary strings with no two consecutive 1s and S n (T 3). In 2005, Egge and Mansour generalized the first Simion-Simion’s result and showed that S n (T p ), the set of permutations avoiding the patterns T p ={12…p, 132, 213}, is counted by the (p–1)th order Fibonacci numbers. In this paper we extend the second Simion-Schmidt’s result by giving a bijection between the set of length (n–1) binary strings with no (p–1) consec…

Classical sequences revisited with permutations avoiding dotted pattern

2011

International audience; Inspired by the definition of the barred pattern-avoiding permutation, we introduce the new concept of dotted pattern for permutations. We investigate permutations classes avoiding dotted patterns of length at most 3, possibly along with other classical patterns. We deduce some enumerating results which allow us to exhibit new families of permutations counted by the classical sequences: 2^n, Catalan, Motzkin, Pell, Fibonacci, Fine, Riordan, Padovan, Eulerian.

Circular sturmian words and Hopcroft’s algorithm

2009

AbstractIn order to analyze some extremal cases of Hopcroft’s algorithm, we investigate the relationships between the combinatorial properties of a circular sturmian word (x) and the run of the algorithm on the cyclic automaton Ax associated to (x). The combinatorial properties of words taken into account make use of sturmian morphisms and give rise to the notion of reduction tree of a circular sturmian word. We prove that the shape of this tree uniquely characterizes the word itself. The properties of the run of Hopcroft’s algorithm are expressed in terms of the derivation tree of the automaton, which is a tree that represents the refinement process that, in the execution of Hopcroft’s alg…

Factorizations of the Fibonacci Infinite Word

2015

The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers.