Basic Combinatorics.

Posted in Uncategorized by pgadey on 2014/07/27

First we recall a little bit of terminology:

1.1. Sets and functions

A set is a collection of elements . We write a set by surrounding its list elements with curly braces. For example: ${X = \{1,2,3\}}$, ${Y = \{\heartsuit, \clubsuit, \star\}}$. We also use set constructor notation ${Y = \{x : P(x)\}}$ where ${P(x)}$ is some statement about ${x}$ that can be true or false. For example: ${X = \{n : n\text{ is even}\}}$, ${Z = \{n : n\text{ is prime}\}}$. We write: ${\{\} = \emptyset}$, ${{\mathbb N}}$ for the set of natural numbers, ${{\mathbb Q}}$ for the set of rational numbers, ${{\mathbb Z}}$ for the set of integers.

We write ${x \in X}$ to mean that ${x}$ is in the set ${X}$. We write ${X \cup Y = \{x : x \in X \text{ or } x \in Y\}}$. We write ${X \cap Y = \{x : x \in X \text{ and } x \in Y\}}$. We write ${X \sqcup Y}$ for ${X \cup Y}$ if ${X \cap Y = \emptyset}$. If ${X \cap Y = \emptyset}$ then we say that ${X}$ and ${Y}$ are disjoint sets.

We write ${X \times Y = \{(x,y) : x \in X,\ y \in Y\}}$ for the set of ordered pairs of elements.

Definition 1 A function ${f : X \rightarrow Y}$ is injective (one to one) if: ${x \neq y}$ implies ${f(x) \neq f(y)}$. A function is surjective (onto) if: for all ${y \in Y}$ there is ${x \in X}$ such that ${f(x) = y}$. A function is bijective if: for all ${y \in Y}$ there is ${x \in X}$ such that ${f(x) = y}$. The number of elements in a set ${X}$ is written ${|X|}$.

1.2. Basic formulae

The basic facts of combinatorics are very simple.

1. If ${k < n}$ then there is no injective function from a set with ${n}$ elements to a set with ${k}$ elements. (This is called pigeon hole principle.)
2. If there is a bijective function from ${X}$ to ${Y}$ then ${|X| = |Y|}$.
3. If ${X}$ and ${Y}$ are disjoint then ${|X \cup Y| = |X| + |Y| - |X \cap Y|}$.
4. If ${X}$ and ${Y}$ are disjoint then ${|X \sqcup Y| = |X| + |Y|}$.
5. ${|X \times Y| = |X| \cdot |Y|}$.
6. The number of ${k}$ element subsets of a set with ${n}$ elements is: ${\binom{n}{k} = \frac{ n! }{ (n-k)! k! }}$. (Why is this an integer? Prove it.)
7. The number of functions from an ${n}$-element set to a ${k}$ element set is ${k^n}$.
8. If ${|X| = n}$ then the number of bijective functions from ${X}$ to ${X}$ (permutations of ${X}$) is ${n!}$.

There are a couple formulae that are handy to remember:

1. There are ${2^n}$ subset of ${\{1, \dots, n\}}$.
2. Suppose you have ${n_1, \dots, n_k}$ objects of types 1, 2, ${\dots}$, ${k}$ respectively. The number of ways of arranging all the objects is: ${{ (n_1 + \dots + n_k)! }/{ n_1! n_2! \dots n_k! }}$
3. Suppose you have ${n}$ identical objects that you want to distribute among ${k}$ containers. The number of ways to do this is: ${\binom{n+k-1}{k-1}}$. (Why?)

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Of Committees and Meetings

Posted in Uncategorized by pgadey on 2014/04/07

I’m happy to announce that I survived my PhD supervisory committee meeting. After 362 days, my committee met once again (we last met Apr 10th 2013), and things went well.

Below I’ve included some details on the process and some post-meeting thoughts. All of it is based purely on my own experience, but might be helpful to others.

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2014 Mentoring

Posted in Math by pgadey on 2014/02/08

I’ve added a page about the mentoring project that I’m working on with three Gr. 12 students this semester. I’ll be updating the page as we go. For more information see Mentoring — 2014. From the introductory remarks:

The plan for the semester is an ambitious one. We’re going to understand the structure of all the regular convex polytopes in all dimensions, and build up a intuition for dimensions greater than three. We’ll spend most of our time learning the tools we need to understand how a geometric object can be pieced together. These tools will include vectors, metric spaces, symmetry groups, and simplicial complexes. I’m taking a very broad view of what constitutes a tool and counts as information about a space. The final result of the project will be a poster presentation about the solids and some 3-dimensional “nets” of the 4-dimemsional solids (the simplex, cube, cross-polytope, and 24-cell).

I can’t resist saying things about hyperbolic geometry. Therefore, if we get to the end of the proposed project, we’ll take a stab at using out high brow high dimensional intuition to understand the Gieseking manifold. There is more than enough stuff to say about the platonic solids, so we’ll see how far we get.

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The Rotationally Distinct Ways to Label a Die

Posted in Math by pgadey on 2013/07/24

I’m giving a talk at the Canadian Math Camp this year. I’ll be showing the kids of how to count the number of ways to label a six sided die up to the rotational symmetries of the cube. Here is the handout for the talk with questions about dice labellings, the 15-puzzle, and permutation groups.

For the curious the labellings are below the cut. Please note that there are typos in the table below. Alex Fink kindly pointed them out and they will be fixed eventually. For now they are an exercise in keen observation.

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Posted in Math by pgadey on 2013/07/20

Over the past couple weeks I’ve been asked a lot of questions about discs in Euclidean space. In this post we’ll be putting pennies on a table, refining covers of discs, and trying to cram lots of balls into high dimensional balls. Some open questions about putting pennies on tables occur below.

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A Bijection

Posted in Math by pgadey on 2013/03/30

While grading an assignment on cardinality, I ran into the answer to the following problem:

Exercise 1 Show that ${f(n) = \sum_{k=0}^n (-1)^{k+1} k}$ is a bijective map ${{\mathbb N} \cup \{0\} \rightarrow {\mathbb Z}}$.

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The Spherical Isoperimetric Inequality

Posted in Math by pgadey on 2013/03/28

Here is an application of the spherical isoperimetric inequality.

Fact You can’t cut up a beach ball into equal parts with a path that is too short.

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Antipodal points after Vîlcu

Posted in Math by pgadey on 2013/03/26

I’ve been thinking a lot about convex bodies in ${{\mathbb R}^3}$ lately. This post is going to be a write up of a useful lemma in the paper: Vîlcu, Constin, On Two Conjectures of Steinhaus, Geom. Dedicata 79 (2000), 267-275.

Let ${S}$ be a centrally symmetric convex body in ${{\mathbb R}^3}$. Let ${d(x,y)}$ denote thes intrinsic metric of ${S}$ and ${D = \sup_{x,y} d(x,y)}$ its intrinsic diameter. For a point ${x \in S}$ we write ${\bar{x}}$ for its image under the central symmetry.

Lemma 1 (Vîlcu) If ${d(x,y) = D}$ then ${y = \bar{x}}$.

This lemma says that if a pair realizes the inner diameter of a centrally symmetric convex body, the pair has to be centrally symmetric. This aligns well with our intuition about the sphere and cube, for example.

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Group Theory Problems

Posted in Math by pgadey on 2013/03/26

Mike Pawliuk and I got talking about elementary group theory problems today. I wanted to record one of my favourites. I heard this one from Lucy Kadets, who heard it from Yuri Burda. I’m not sure if he is the original author or not.

Let ${S_n}$ denote the symmetric group on ${n}$ elements. We say ${H \leq S_n}$ is a point-fixing subgroup if there is a ${1 \leq k \leq n}$ such that ${h(k) = k}$ for all ${h \in H}$. We say ${H}$ has fixed points if for each ${h \in H}$ there is ${1 \leq k_h \leq n}$ such that ${h(k_h) = k_h}$.

Exercise 1 Is every ${H \leq S_n}$ that has fixed points a point-fixing subgroup?

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Polynomials

Posted in Math by pgadey on 2013/03/01

Brandon Hanson told me the following elementary number theory problems last night.

Exercise 1 Every non-constant polynomial takes on a composite value.

Hint: Look at ${f(x) = p}$ and ${f(kp + x)}$.

Exercise 2 If a non-constant polynomial takes on infinitely many prime values then it is irreducible.