Related papers: Exactly solved models of polyominoes and polygons
In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of…
Hexagonal polyominoes are polyominoes on the honeycomb lattice. We enumerate the symmetry classes of convex hexagonal polyominoes. Here convexity is to be understood as convexity along the three main column directions. We deduce the…
This paper concerns the enumeration of rotation-type and congruence-type convex polyominoes on the square lattice. These can be defined as orbits of the groups C4, of rotations, and D4, of symmetries of the square acting on (translation-…
An alternative generating function is proposed to enumerate row-convex polyominoes without internal holes on a discrete grid. The approach is based on integer partitions of the total area, where each partition corresponds to a sequence of…
Our goal in this work is to found a closed form for rational generat- ing functions, these generate a various families of polynomials and generalized polynomials, in order to get the general recursive formula satisfied by these polynomials.
Convex polyominoes can be refined according to the number of direction changes in monotone paths connecting pairs of cells, leading to the notion of $k$-convexity. In particular, the cases $k=1$ and $k=2$ correspond to $L$-convex and…
Back in the early days of polyomino enumeration, a model called column-convex polyominoes was introduced and its area generating function was found. That generating function is rational: the numerator has degree four and the denominator has…
In most of today's exactly solved classes of polyominoes, either all members are convex (in some way), or all members are directed, or both. If the class is neither convex nor directed, the exact solution uses to be elusive. This paper is…
In this paper, we consider the problem of avoiding a single vincular pattern of length three by derangements in the flattened sense and find explicit formulas for the generating functions enumerating members of each corresponding avoidance…
We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…
We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study…
For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
The aim of this work is the study of the class of periodic parallelogram polyominoes, and two of its variantes. These objets are related to 321-avoiding affine permutations. We first provide a bijection with the set of triangles under Dyck…
The main goal in this manuscript is to present a class of functions satisfying a certain orthogonality property for which there also exists a three term recurrence formula. This class of functions, which can be considered as an extension to…
Column-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating function. In this chapter we generalize that result. Let a p-column polyomino be a polyomino whose columns…
Every classical orthogonal polynomial system $p_n(x)$ satisfies a three-term recurrence relation of the type \[ p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x)~ (n=0,1,2,\ldots, p_{-1}\equiv 0), \] with $C_nA_nA_{n-1}>0$. Moreover, Favard's…
The method of constructing spline classes in the form of trigonometric Fourier series whose coefficients have a certain decreasing order are considered. in turn, this decrement determines the number of continuous derivatives of sum of this…
Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…
We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…