The “Fibonacci Dichotomy” of Kaiser and Klazar [15] was one of the first general results on the enumeration of permutation classes. It states that if there are fewer permutations of length n in a permutation class than the nth Fibonacci number, for any n, then the enumeration of the class is given by a polynomial for sufficiently large n. Since the Fibonacci Dichotomy was established for permut...

The Fibonacci Hypercube is defined as the polytope determined by the convex hull of the “Fibonacci” strings, i.e., binary strings of length n having no consecutive ones. We obtain an efficient characterization of vertex adjacency and use this to study the graph of the Fibonacci Hypercube. In particular we discuss a decomposition of the graph into self-similar subgraphs that are also graphs of F...

Abstract. We study the statistics of column-convex lattice animals resulting from the stacking of squares on a single or double staircase. We obtain exact expressions for the number of animals with a given length and area, their mean length and their mean height. These objects are closely related to Fibonacci numbers. On a single staircase, the total number of animals with area k is given by th...

Zeckendorf’s theorem states that every positive integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers. This theorem induces a binary numeration system for the positive integers known as Fibonacci coding. Fibonacci code is a variable-length prefix code that is robust against insertion and deletion errors and is useful in data transmission and data compression. In this pape...

Data compression has been widely applied in many data processing areas. Compression methods use variable-length codes with the shorter codes assigned to symbols or groups of symbols that appear in the data frequently. There exist many coding algorithms, e.g. Elias-delta codes, Fibonacci codes and other variable-length codes which are often applied to encoding of numbers. Although we often do no...

Generalized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and nonembeddable infinite series are given. The question is completely solved for strings f of leng...

We show that the p-adic valuation of the sequence of Fibonacci numbers is a p-regular sequence for every prime p. For p 6= 2, 5, we determine that the rank of this sequence is α(p) + 1, where α(m) is the restricted period length of the Fibonacci sequence modulo m.

We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length n in a permutation class is either at least as large as the nth Fibonacci number or is eventually polynomial.

Recent publications advocate the use of various variable length codes for which each codeword consists of an integral number of bytes in compression applications using large alphabets. This paper shows that another tradeoff with similar properties can be obtained by Fibonacci codes. These are fixed codeword sets, using binary representations of integers based on Fibonacci numbers of order m ≥ 2...

In this paper, we determine the maximum number of distinct Lyndon factors that a word of length n can contain. We also derive formulas for the expected total number of Lyndon factors in a word of length n on an alphabet of size σ, as well as the expected number of distinct Lyndon factors in such a word. The minimum number of distinct Lyndon factors in a word of length n is 1 and the minimum tot...