## Ordinary Generating Functions and Recurrence Equations

1) Ordinary generating functions of a variable Generating functions are an important tool for solving combinatorial problems of various types. A typical problem is the counting of the number of objects as a function of the size $$n$$, which we can denote by $$a_{n}$$. Thus, for each value Read more…

## Iterated Function Systems, Fractals and Sierpinski Triangle

In a previous article we have introduced some examples of fractals, illustrating their main characteristics, both qualitative and quantitative: self-similarity, geometric irregularity, fractional dimension. To continue the study of fractal science, it’s first necessary to give a more rigorous definition of the mathematical context in which fractal objects are defined.Each Read more…

## Dirichlet’s Box Principle and Ramsey Numbers

The box principle is attributed to the German mathematician Dirichlet (1805-1859). It is also called the pigeonhole principle. This article illustrates Dirichlet’s principle and gives a brief introduction to Ramsey’s theory, with some examples of computation of Ramsey numbers. For a deeper understanding of the topics of this article you Read more…

## Cardano, Gambling and the dawn of Probability Theory

The birth of probability theory is usually set in the mid-seventeenth century. At that time the two great mathematicians Blaise Pascal (1623–1662) and Pierre de Fermat (1601– 1665) discussed together some gambling problems and defined the theoretical basis of the mathematical theory of classical probability.However, the first studies on the Read more…

## The Principle of Inclusion-Exclusion

The principle of inclusion-exclusion is an important result of combinatorial calculus which finds applications in various fields, from Number Theory to Probability, Measurement Theory and others. In this article we consider different formulations of the principle, followed by some applications and exercises. For further information, see for example [1] and Read more…

## Euler’s Polygon Triangulation Problem

Problem Let P be a convex polygon with $$n$$ sides. Calculate in how many different ways the polygon can be divided into triangles using diagonals that do not intersect each other in the interior of P. This problem was proposed by Euler in 1751 to his friend Christian Goldbach. Read more…

## Counting Numbers with Adjacent Digits

Problem Suppose we only use the digits of the set $$A = \{1,2,3,4,5 \}$$. How many numbers of $$n$$ digits can be formed with the set $$A$$, if all adjacent digits differ exactly by $$1$$? We denote the number we are looking for with $$a(n)$$. HintIf \(n = Read more…

## Introduction to Fractals – Koch Snowflake

Euclidean geometry studies geometric objects such as lines, triangles, rectangles, circles, etc. Fractals are also geometric objects; however, they have specific properties that distinguish them and cannot be classified as objects of classical geometry. Although Mandelbrot (1924-2010) is generally considered the father of the scientific theory of fractals, in reality the ideas underlying Read more…