National Research University Higher School of Economics

Об этом курсе

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.

Section

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1 video (Total 8 min), 4 readings

Course Overview10m

Grading and Logistics10m

Suggested Readings0m

About the Instructor10m

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 (Total 78 min), 1 quiz

Words9m

Permutations10m

k-permutations8m

Merry-go-rounds and Fermat’s little theorem 18m

Merry-go-rounds and Fermat’s little theorem 211m

Binomial coefficients14m

The Pascal triangle16m

Quiz 20m

Section

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 (Total 87 min), 1 quiz

Balls in boxes and multisets 110m

Balls in boxes and multisets 26m

Integer compositions11m

Principle of inclusion and exclusion: two examples12m

Principle of inclusion and exclusion: general statement9m

The derangement problem19m

Quiz 30m

Section

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 (Total 105 min), 1 reading, 1 quiz

Fibonacci numbers and the Pascal triangle7m

Domino tilings8m

Vending machine problem10m

Linear recurrence relations: definition7m

The characteristic equation8m

Linear recurrence relations of order 211m

The Binet formula11m

Sidebar: the golden ratio9m

Linear recurrence relations of arbitrary order8m

The case of roots with multiplicities12m

Spoilers! Solutions for quizzes 2, 3, and 4.0m

Quiz 40m

Section

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 (Total 73 min), 1 reading, 1 quiz

Recurrence relation for triangulations11m

The cashier problem9m

Dyck paths5m

Recurrence relations for Dyck paths9m

Reflection trick and a formula for Catalan numbers12m

Binary trees15m

Solutions10m

Section

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 (Total 87 min), 1 reading, 1 quiz

Formal power series11m

When are formal power series invertible?9m

Derivation of formal power series12m

Binomial theorem for negative integer exponents8m

Solving the Fibonacci recurrence relation9m

Solving the Fibonacci recurrence 2: Binet formula6m

Generating functions of linear recurrence relations are rational7m

Solving linear recurrence relations: general case10m

Math expressions10m

Quiz 60m

Section

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 (Total 61 min), 1 quiz

Derivation and integration of formal power series10m

Chain rule. Inverse function theorem7m

Logarithm. Logarithmic derivative5m

Binomial theorem for arbitrary exponents13m

Generating function for Catalan numbers14m

Quiz 70m

Section

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 (Total 87 min), 1 reading, 1 quiz

Young diagrams4m

Generating function for partitions15m

Partitions with odd and distinct summands11m

Sylvester’s bijection8m

Euler’s pentagonal theorem12m

Proof of Euler’s pentagonal theorem 18m

Proof of Euler’s pentagonal theorem 214m

Computing the number of partitions via the pentagonal theorem6m

Spoilers! Solutions for quizzes 6, 7, and 8.0m

Quiz 80m

Section

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 (Total 80 min), 1 reading, 1 quiz

q-binomial coefficients: definition and first properties10m

Recurrence relation for q-binomial coefficients 114m

Recurrence relation for q-binomial coefficients 23m

Explicit formula for q-binomial coefficients11m

Euler’s partition function8m

Sidebar: q-binomial coefficients in linear algebra9m

q-binomial theorem10m

Solutions10m

4.7

By RA•Mar 30th 2018

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 RR•Aug 22nd 2017

Great lectures and content. I really enjoyed it. However, the solutions exercises could be clearer and in more detail. Thank you!

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