# Introduction to Enumerative Combinatorics

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## Introduction to Enumerative Combinatorics

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### Beschrijving

About this course: Enumerative combinatorics deals with finite sets and their cardinalities. In other words, a typical problem of enumerative combinatorics is to find the number of ways a certain pattern can be formed. In the first part of our course we will be dealing with elementary combinatorial objects and notions: permutations, combinations, compositions, Fibonacci and Catalan numbers etc. In the second part of the course we introduce the notion of generating functions and use it to study recurrence relations and partition numbers. The course is mostly self-contained. However, some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very h…

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About this course: Enumerative combinatorics deals with finite sets and their cardinalities. In other words, a typical problem of enumerative combinatorics is to find the number of ways a certain pattern can be formed. In the first part of our course we will be dealing with elementary combinatorial objects and notions: permutations, combinations, compositions, Fibonacci and Catalan numbers etc. In the second part of the course we introduce the notion of generating functions and use it to study recurrence relations and partition numbers. The course is mostly self-contained. However, some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very helpful.

Who is this class for: This course is primarily aimed at the Master of Science students of Mathematics department. Students are expected to have knowledge of one-semester courses of advanced calculus and linear algebra.

Created by:  Higher School of Economics
• Taught by:  Evgeny Smirnov, Associate Professor

Faculty of Mathematics
Level Intermediate Commitment 8 weeks Language English How To Pass Pass all graded assignments to complete the course. User Ratings 4.6 stars Average User Rating 4.6See what learners said Coursework

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Higher School of Economics National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru

Syllabus

WEEK 1

Introduction

1. Video: Introduction

Permutations and binomial coefficients

In this introductory lecture we discuss fundamental combinatorial constructions: we will see how to compute the number of words of fixed length in a given alphabet, the number of permutations of a finite set and the number of subsets with a given number of elements in a finite set. The latter numbers are called binomial coefficients; we will see how they appear in various combinatorial problems in this and forthcoming lectures. As an application of combinatorial methods, we also give a combinatorial proof of Fermat's little theorem.

7 videos expand

1. Video: Words
2. Video: Permutations
3. Video: k-permutations
4. Video: Merry-go-rounds and Fermat’s little theorem 1
5. Video: Merry-go-rounds and Fermat’s little theorem 2
6. Video: Binomial coefficients
7. Video: The Pascal triangle

WEEK 2

Binomial coefficients, continued. Inclusion and exclusion formula.

In the first part of this lecture we will see more applications of binomial coefficients, in particular, their appearance in counting multisets. The second part is devoted to the principle of inclusion and exclusion: a technique which allows us to find the number of elements in the union of several sets, given the cardinalities of all of their intersections. We discuss its applications to various combinatorial problem, including the computation of the number of permutations without fixed points (the derangement problem).

7 videos expand

1. Video: The cab driver problem
2. Video: Balls in boxes and multisets 1
3. Video: Balls in boxes and multisets 2
4. Video: Integer compositions
5. Video: Principle of inclusion and exclusion: two examples
6. Video: Principle of inclusion and exclusion: general statement
7. Video: The derangement problem

WEEK 3

Linear recurrences. The Fibonacci sequence
We start with a well-known "rabbit problem", which dates back to Fibonacci. Using the Fibonacci sequence as our main example, we discuss a general method of solving linear recurrences with constant coefficients.

1. Video: Fibonacci’s rabbit problem
2. Video: Fibonacci numbers and the Pascal triangle
3. Video: Domino tilings
4. Video: Vending machine problem
5. Video: Linear recurrence relations: definition
6. Video: The characteristic equation
7. Video: Linear recurrence relations of order 2
8. Video: The Binet formula
9. Video: Sidebar: the golden ratio
10. Video: Linear recurrence relations of arbitrary order
11. Video: The case of roots with multiplicities
12. Reading: Spoilers! Solutions for quizzes 2, 3, and 4.

WEEK 4

A nonlinear recurrence: many faces of Catalan numbers
In this lecture we introduce Catalan numbers and discuss several ways to define them: via triangulations of a polygon, Dyck paths and binary trees. We also prove an explicit formula for Catalan numbers.

1. Video: Triangulations of a polygon
2. Video: Recurrence relation for triangulations
3. Video: The cashier problem
4. Video: Dyck paths
5. Video: Recurrence relations for Dyck paths
6. Video: Reflection trick and a formula for Catalan numbers
7. Video: Binary trees

WEEK 5

Generating functions: a unified approach to combinatorial problems. Solving linear recurrences
We introduce the central notion of our course, the notion of a generating function. We start with studying properties of formal power series and then apply the machinery of generating functions to solving linear recurrence relations.

9 videos expand

1. Video: Generating functions: first examples
2. Video: Formal power series
3. Video: When are formal power series invertible?
4. Video: Derivation of formal power series
5. Video: Binomial theorem for negative integer exponents
6. Video: Solving the Fibonacci recurrence relation
7. Video: Solving the Fibonacci recurrence 2: Binet formula
8. Video: Generating functions of linear recurrence relations are rational
9. Video: Solving linear recurrence relations: general case

WEEK 6

Generating functions, continued. Generating function of the Catalan sequence

In this lecture we discuss further properties of formal power series. In particular, we prove an analogue of the binomial theorem for an arbitrary rational exponent. We apply this technique to computing the generating function of the sequence of Catalan numbers.

6 videos expand

1. Video: Composition of formal power series
2. Video: Derivation and integration of formal power series
3. Video: Chain rule. Inverse function theorem
4. Video: Logarithm. Logarithmic derivative
5. Video: Binomial theorem for arbitrary exponents
6. Video: Generating function for Catalan numbers

WEEK 7

Partitions. Euler’s generating function for partitions and pentagonal formula

In this lecture we introduce partitions, i.e. the number of ways to present a given integer as a sum of ordered integer summands. There is no closed formula for the number of partitions; however, it is possible to compute their generating function. We study the properties of this generating function, including the famous Pentagonal theorem, due to Leonhard Euler.

1. Video: Definition and first examples
2. Video: Young diagrams
3. Video: Generating function for partitions
4. Video: Partitions with odd and distinct summands
5. Video: Sylvester’s bijection
6. Video: Euler’s pentagonal theorem
7. Video: Proof of Euler’s pentagonal theorem 1
8. Video: Proof of Euler’s pentagonal theorem 2
9. Video: Computing the number of partitions via the pentagonal theorem
10. Reading: Spoilers! Solutions for quizzes 6, 7, and 8.

WEEK 8

Gaussian binomial coefficients. “Quantum” versions of combinatorial identities

Our final lecture is devoted to the so-called "q-analogues" of various combinatorial notions and identities. As a general principle, we replace identities with numbers by identities with polynomials in a certain variable, usually denoted by q, that return the original statement as q tends to 1. This approach turns out to be extremely useful in various branches of mathematics, from number theory to representation theory.

1. Video: Generating function for partitions inside a rectangle
2. Video: q-binomial coefficients: definition and first properties
3. Video: Recurrence relation for q-binomial coefficients 1
4. Video: Recurrence relation for q-binomial coefficients 2
5. Video: Explicit formula for q-binomial coefficients
6. Video: Euler’s partition function
7. Video: Sidebar: q-binomial coefficients in linear algebra
8. Video: q-binomial theorem

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