## Combinatorics (I)

This book is not a particularly easy read, compared to what is the general format of the series in which it is published, but this is a good thing in my view as it also means the author managed to go into enough details in specific contexts to touch upon at least some properties/topics of interest. You don’t need any specific background knowledge to read and understand the book – at least not any sort of background knowledge one would not expect someone who might decide to read a book like this one to already have – but you do need when reading it to have the sort of mental surplus that enables you to think carefully about what’s going on and devote a few mental resources to understanding the details.

Some quotes and links from the first half of the book below.

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“The subject of *combinatorial analysis* or *combinatorics* […] [w]e may loosely describe [as] the branch of mathematics concerned with *selecting*, *arranging*, *constructing*, *classifying*, and *counting *or *listing *things. […] the subject involves *finite sets* or *discrete elements* that proceed in separate steps […] rather than continuous systems […] Mathematicians sometimes use the term ‘combinatorics’ to refer to a larger subset of discrete mathematics that includes graph theory. In that case, what is commonly called combinatorics is then referred to as ‘enumeration’. […] Combinatorics now includes a wide range of topics, some of which we cover in this book, such as the geometry of tilings and polyhedra […], the theory of graphs […], magic squares and latin squares […], block designs and finite projective planes […], and partitions of numbers […]. [The] chapters [of the book] are largely independent of each other and can be read in any order. Much of combinatorics originated in recreational pastimes […] in recent years the subject has developed in depth and variety and has increasingly become a part of mainstream mathematics. […] Undoubtedly part of the reason for the subject’s recent importance has arisen from the growth of computer science and the increasing use of algorithmic methods for solving real-world practical problems. These have led to combinatorial applications in a wide range of subject areas, both within and outside mathematics, including network analysis, coding theory, probability, virology, experimental design, scheduling, and operations research.”

“[C]ombinatorics is primarily concerned with four types of problem:

*Existence problem*: Does □□□ exist?

*Construction problem*: If □□□ exists, how can we construct it?

*Enumeration problem*: How many □□□ are there?

*Optimization problem*: Which □□□ is best? […]

[T]hese types of problems are not unrelated; for example, the easiest way to prove that something exists may be to construct it explicitly.”

“In this book we consider two types of enumeration problem – *counting problems *in which we simply wish to know the number of objects involved, and *listing problems* in which we want to list them all explicitly. […] It’s useful to have some basic counting rules […] In what follows, all the sets are finite. […] In general we have the following rule; here, subsets are disjoint if they have no objects in common: *Addition rule:* To find the number of objects in a set, split the set into disjoint subsets, count the objects in each subset, and add the results. […] *Subtraction rule:* If a set of objects can be split into two subsets A and B, then the number of objects in *B* is obtained by subtracting the number of objects in A from the number in the whole set. […] The subtraction rule extends easily to sets that are split into more than two subsets with no elements in common. […] the *inclusion-exclusion principle* […] extends this simple idea to the situation where the subsets may have objects in common. […] In general we have the following result: *Multiplication rule:* If a counting problem can be split into stages with several options at each stage, then the total number of possibilities is the product of options at each stage. […] Another useful principle in combinatorics is the following: *Correspondence rule:* We can solve a counting problem if we can put the objects to be counted in one-to-one correspondence with the objects of a set that we have already counted. […] We conclude this section with one more rule: *Division rule:* If a set of *n* elements can be split into *m* disjoint subsets, each of size *k*, then *m = n / k*.”

“Every algorithm has a *running time *[…] this may be the time that a computer needs to carry out all the necessary calculations, or the actual number of such calculations. Each problem [also] has an *input size* […] the running time *T *usually depends on the input size *n*. Particularly important, because they’re the most efficient, are the *polynomial-time algorithms*, where the maximum running time is proportional to a power of the input size […] The collection of all polynomial-time algorithms is called P. […] In contrast, there are inefficient algorithms that don’t take polynomial time, such as the exponential-time algorithms […] At this point we introduce NP, the set of ‘non-deterministic polynomial-time problems’. These are algorithms for which a solution, when given, can be *checked* in polynomial time. Clearly P is contained in NP, since if a problem can be solved in polynomial time then a solution can certainly be checked in polynomial time – checking solutions is far easier than finding them in the first place. But are they the same? […] Few people people believe that the answer is ‘yes’, but no one has been able to prove that P ≠ NP. […] a problem is *NP-complete *if its solution in polynomial time means that *every* NP problem can be solved in polynomial time. […] If there were a polynomial algorithm for just one of them, then polynomial algorithms would exist for the whole lot and P would equal NP. On the other hand, if just one of them has no polynomial algorithm, then none of the others could have a polynomial algorithm either, and P would be different from NP.”

“In how many different ways can *n* objects be arranged? […] generally, we have the following result: *Arrangements*: The number of arrangements of *n *objects is n x (n -1) x (n – 2) x … x 3 x 2 x 1. This number is called *n factorial* and is denoted by *n!*. […] The word permutation is used in different ways. We’ll use it to mean an *ordered selection without repetition*, while others may use it to mean an arrangement […] generally, we have the following rule: *Ordered selections without repetition (permutations)*: If we select *k* items from a set of *n* objects, and if the selections are ordered and repetition is not allowed, then the number of possible selections is *n x (n – 1) x (n – 2) x … x (n – k +1)*. We denote this expression by *P(n,k)*. […] Since *P(n, n) = n x (n -1) x (n – 2) x … x 3 x 2 x 1 = n!*, an arrangement is a permutation for which *k = n*. […] generally, we have the following result: *P(n,k) = n! /(n-k)!*. […] unordered selections without repetition are called *combinations*, giving rise to the words *combinatorial* and *combinatorics*. […] generally, we have the following result: *Unordered selections without repetition (combinations)*: If we select *k *items from a set of *n* objects, and if the selections are unordered and repetition is not allowed, then the number of possible selections is *P(n,k)/k! = n x (n-1) x (n-2) x … x (n – k + 1)/k!*. We denote this expression by *C(n,k)* […] *Unordered selections with repetition*: If we select *k *items from a set of *n* objects, and if the selections are unordered and repetition is allowed, then the number of possible selections is C(n + *k *– 1, *k*). […] *Combination rule 1*: For any numbers *k* and *n* with *k ≤ n, C(n,k) = C(n,n-k)* […] *Combination rule 2*: For any numbers *n* and *k* with *k ≤ n*, *C(n, n-k) = n!/(n-k)!(n-(n-k))! = n!/(n-k)!k! = C(n,k). *[…] *Combination rule 3*: For any number *n, C(n,0) + C(n,1) + C(n,2) + … + C(n,n-1) + C(n,n) =* *2 ^{n}*”

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Links:

Tilings/Tessellation.

Knight’s tour.

Seven Bridges of Königsberg problem.

Three utilities problem.

Four color theorem.

Tarry’s algorithm (p.7) (formulated slightly differently in the book, but it’s the same algorithm).

Polyomino.

Arthur Cayley.

Combinatorial principles.

Minimum connector problem.

Travelling salesman problem.

Algorithmic efficiency. Running time/time complexity.

Boolean satisfiability problem. Cook–Levin theorem.

Combination.

Mersenne primes.

Permutation. Factorial. Stirling’s formula.

Birthday problem.

Varāhamihira.

Manhattan distance.

Fibonacci number.

Pascal’s triangle. Binomial coefficient. Binomial theorem.

Pigeonhole principle.

Venn diagram.

Derangement (combinatorial mathematics).

Tower of Hanoi.

Stable marriage problem. Transversal (combinatorics). Hall’s marriage theorem.

Generating function (*the topic covered in the book more specifically is related to a symbolic generator of the subsets of a set, but a brief search yielded no good links to this particular topic – US*).

Group theory.

Ferdinand Frobenius. Burnside’s lemma.

You can also read Pranav Sriram’s book if you want a motivating textbook that also teaches.

Other books include Richard Stan’s book on enumerative combinatorics, and additive combinatorics (I was recently reading).

On my blog I recently wrote a post relevant to the Sylvester Gallai problem which is combinatorical and so you may like it.

Comment by adityaguharoy | August 5, 2018 |