Related papers: Statistics on Almost-Fibonacci Pattern-Avoiding Pe…

For each positive integer $n$, the Fibonacci-sum graph $G_n$ on vertices $1,2,\ldots,n$ is defined by two vertices forming an edge if and only if they sum to a Fibonacci number. It is known that each $G_n$ is bipartite, and all Hamiltonian…

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that \begin{equation*} \log \operatorname{lcm} (F_1, F_2, \dots, F_n) \sim \frac{3 \log \alpha}{\pi^2} \cdot n^2 \quad \text{as } n \to +\infty,…

There are three long-known types of restricted integer compositions whose counts match the Fibonacci sequence:\ one from ancient India and two from 19th century England. We give proofs of these enumeration results using tiling arguments and…

Denote by $A_n$ the set of square $(0,1)$ matrices of order $n$. The set $A_n$, $n\le8$, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of…

We compute the Frobenius number for numerical semigroups generated by the squares of three consecutive Fibonacci numbers. We achieve this by using and comparing three distinct algorithmic approaches: those developed by Ram\'irez Alfons\'in…

The maximally clustered permutations are characterized by avoiding the classical permutation patterns 3421, 4312, and 4321. This class contains the freely-braided permutations and the fully-commutative permutations. In this work, we show…

For $\tau\in S_3$, let $S_n(\tau)$ denote the set of permutations in $S_n$ which avoid the pattern $\tau$, and let $E_n^\tau$ denote the expectation with respect to the uniformly random probability measure on $S_n(\tau)$. Let…

We motivate the study of a certain class of random Fibonacci sequences - which we call continuous random Fibonacci sequences - by demonstrating that their exponential growth rate can be used to establish capacity and power scaling laws for…

Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234-avoiding permutations,…

The convolved Fibonacci numbers F_j^(r) are defined by (1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^(r)z^j. In this note some related numbers that can be expressed in terms of convolved Fibonacci numbers are considered. These numbers appear in the…

In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of $n$ 3-cycles…

We show that the number of signed permutations avoiding 1234 equals the number of signed permutations avoiding 2143 (also called vexillary signed permutations), resolving a conjecture by Anderson and Fulton. The main tool that we use is the…

The Fibonacci sequence is a series of positive integers in which, starting from $0$ and $1$, every number is the sum of two previous numbers, and the limiting ratio of any two consecutive numbers of this sequence is called the golden ratio.…

We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares…

A permutation of the integers avoiding monotone arithmetic progressions of length $6$ was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length…

The Fibonacci sequence, $F_n = F_{n - 1} + F_{n - 2}$, and its counterpart for $n < 0$, the negaFibonacci sequence, $F_{-n} = (-1)^{n + 1} \cdot F_n$, are among the most studied sequences in mathematics. In this paper we will present a new…

Let $f_4(n)$ be the number of hyperquaternary representations of $n$ and $b_4(n)$ be the number of balanced quaternary representations of $n$. We show that there is no integer $k$ such that $f_4(n+k)=b_4(n)$ for all $n\ge -k$, in contrast…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n \geq 0$. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we…

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence…

The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice…