Related papers: Generating Trees and Fibonacci Polyominoes

Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…

We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kind of trees or forests, and some more general classes of graphs, ranging from the Connes-Kreimer algebra to an algebra of labelled forests…

We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference…

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of $\omega$-tree languages derived from weak…

In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.

A Catalan word of length $n$ that avoids the pattern $(\geq, \geq)$ is a sequence $w=w_1\cdots w_n$ with $w_1=0$ and $0\leq w_i\leq w_{i-1}+1$ for all $i$, while ensuring that no subsequence satisfies $w_i \geq w_{i+1}\geq w_{i+2}$ for…

The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A…

Given an infinite word, enumerating its factors is an important exercise for understanding the structure of the word. The process of finding all the factors is quite tricky for two-dimensional words. In this paper, two possible ways of…

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the…

We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…

Fibonomial coefficients count the number of specific finite birth self-similar subposets of an infinite non-tree poset naturally related to the Fibonacci tree of rabbits growth process.

Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros conjectured that n_g possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree.…

In this paper we study substitutions and some of their associated generating functions. This association takes aperiodicity to transcendence, and vice-versa. These generating functions have a recursive structure arising from the…

Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…

The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive $1$s. Recently Jianxin Wei and Yujun Yang introduced a one parameter generalization, Fibonacci $p$-cubes $\Gamma_n^p$, which are…

In this paper we consider a restricted class of convex polyominoes that we call Z-convex polyominoes. Z-convex polyominoes are polyominoes such that any two pairs of cells can be connected by a monotone path making at most two turns (like…

In this second paper, we look at the following question: are the properties of the trees associated to the tilings $\{p,4\}$ and $\{p$+$2,3\}$ of the hyperbolic plane still true if we consider a finitely generated tree by the same rules but…

In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these…

In this paper, we provide methods to automatically obtain automata that generate polyominoes inscribed in a rectangle of fixed width and increasing height. We use them to obtain the generating function of those sequences for small widths.

We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial…