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: National Research University Higher School of Economics
Taught by: Evgeny Smirnov, Associate Professor
Faculty of Mathematics
| Level | Intermediate |
| Commitment | 8 weeks, 10-12 hours per week |
| Language | English |
| How To Pass | Pass all graded assignments to complete the course. |
| User Ratings |
Syllabus
WEEK 1
Introduction
1 video, 4 readings
- Reading: Course Overview
- Reading: Grading and Logistics
- Reading: Suggested Readings
- Reading: About the Instructor
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
- Video: Permutations
- Video: k-permutations
- Video: Merry-go-rounds and Fermat’s little theorem 1
- Video: Merry-go-rounds and Fermat’s little theorem 2
- Video: Binomial coefficients
- Video: The Pascal triangle
Graded: Quiz 2
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
- Video: Balls in boxes and multisets 1
- Video: Balls in boxes and multisets 2
- Video: Integer compositions
- Video: Principle of inclusion and exclusion: two examples
- Video: Principle of inclusion and exclusion: general statement
- Video: The derangement problem
Graded: Quiz 3
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.
11 videos, 1 reading
- Video: Fibonacci numbers and the Pascal triangle
- Video: Domino tilings
- Video: Vending machine problem
- Video: Linear recurrence relations: definition
- Video: The characteristic equation
- Video: Linear recurrence relations of order 2
- Video: The Binet formula
- Video: Sidebar: the golden ratio
- Video: Linear recurrence relations of arbitrary order
- Video: The case of roots with multiplicities
- Reading: Spoilers! Solutions for quizzes 2, 3, and 4.
Graded: Quiz 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.
7 videos, 1 reading
- Video: Recurrence relation for triangulations
- Video: The cashier problem
- Video: Dyck paths
- Video: Recurrence relations for Dyck paths
- Video: Reflection trick and a formula for Catalan numbers
- Video: Binary trees
- Reading: Solutions
Graded: Graded assignment 1 (midterm exam)
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, 1 reading
- Video: Formal power series
- Video: When are formal power series invertible?
- Video: Derivation of formal power series
- Video: Binomial theorem for negative integer exponents
- Video: Solving the Fibonacci recurrence relation
- Video: Solving the Fibonacci recurrence 2: Binet formula
- Video: Generating functions of linear recurrence relations are rational
- Video: Solving linear recurrence relations: general case
- Reading: Math expressions
Graded: Quiz 6
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
- Video: Derivation and integration of formal power series
- Video: Chain rule. Inverse function theorem
- Video: Logarithm. Logarithmic derivative
- Video: Binomial theorem for arbitrary exponents
- Video: Generating function for Catalan numbers
Graded: Quiz 7
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.
9 videos, 1 reading
- Video: Young diagrams
- Video: Generating function for partitions
- Video: Partitions with odd and distinct summands
- Video: Sylvester’s bijection
- Video: Euler’s pentagonal theorem
- Video: Proof of Euler’s pentagonal theorem 1
- Video: Proof of Euler’s pentagonal theorem 2
- Video: Computing the number of partitions via the pentagonal theorem
- Reading: Spoilers! Solutions for quizzes 6, 7, and 8.
Graded: Quiz 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.
8 videos, 1 reading
- Video: q-binomial coefficients: definition and first properties
- Video: Recurrence relation for q-binomial coefficients 1
- Video: Recurrence relation for q-binomial coefficients 2
- Video: Explicit formula for q-binomial coefficients
- Video: Euler’s partition function
- Video: Sidebar: q-binomial coefficients in linear algebra
- Video: q-binomial theorem
- Reading: Solutions
Graded: Graded assignment 2 (final exam)
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Ratings and Reviews
Excellent selection of material and presentation; TAs were of great help as well. The techniques taught in this course will be a nice addition to my algorithms analysis toolbox.
by far the best math course I had (and I've taken many) on coursera
I enjoyed the course.
The lecturer was very good and the material quite interesting.
I feel that the assignments where sometimes very different compared to the material covered but I ended up doing extra research and learning more which is good!
Many thanks to the Coursera Staff as well as the Higher School of Economics.
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Funny
Very good presentation