Discrete and Combinatorial Algebra

John Rickert, Associate Professor of Mathematics

Office hours this week: MTRF 8, or make an appointment, or drop in. Here's my schedule

http://www.rose-hulman.edu/~rickert/Classes/ma215

To Homework ...Questions

Homework for our next class ...Today's questions ...Instructions for downloading magma

The factorizations of elements in S

The book is closed book/notes. You are encouraged to bring your computer to aid your calculations.

You may use one page of notes or stored

In the "Questions" section, I'll color the active questions green so that they are easier (in theory) to find as you scan through the page. Please let me know if I've missed anything.

The main goal in this class is to have you (the student) perform as an active learner. To do this you will need to do the exercises, raise questions about structures that you are studying, create hypotheses and test these hypotheses.

The quizzes, examinations and homework done during the year will be worth 80% of the course grade. The final examination will be worth 20% of the grade.

For Friday, 8/31: Read the introduction and section 1.1.1, do exercise 1.0.1 and hand in your definition of shuffled deck.

Your definitions of a shuffled deck have been compiled for your perusal.

Try to download Magma. Instructions for downloading magma are below.

Read Section 1.2. Do exercises 1.1.3 and 1.2.1 Turn in these exercises on Friday, September 7.

Find a convincing explanation for the fact that the number of elements in S

You will find it useful to bring some sort of calculating device (calculator, computer, abacus) to class on Tuesday.

I know that you're all disappointed, but there's nothing to turn in for Thursday's class. We will continue the discussion of exercises 1.1.3 and 1.2.1.

Reread Section 1.3.1 covering perfect riffle shuffles. Experiment a little and see what you can discover.

What is the mathematics behind the card trick?

Reread Section 1.3.1, read Section 1.3.2. Work exercises 1.3.2 and 1.3.3.

Try to see if you can prove that the average number of adjacencies in S

Turn in exercises 1.3.2 and 1.3.3. We will have a quiz covering PRS and TIAR.

Read through exercise 1.4.6.2. Be sure to work exercises 1.4.4, 1.4.5, 1.4.6.1 and 1.4.6.2.

Reread section 1.4 and work enough examples so that you feel comfortable with composition of permutations and the fact that this composition is associative, i.e. (ab)c=a(bc), but not commutative, i.e. ab might not = ba.

Look again at those association schemes and see if you can find a way of determining how many association schemes there are on

Run 100,000,000 random trials using the magma code below. How many times did the pair of elements commute?

Read through exercise 1.4.6.11 and come to class with questions.

If you did not pick up your permutations in class Tuesday, contact me to get your permutations

Read through exercise 1.5.3 - you never know when there might be a pop quiz...

2. Adjacent Trranspositions

3. {tau,rho}- factorization

Exam #1, Friday, September 28. The average score on the exam was 79.4 out of 120.

Read through exercise 1.9.3

Come to class with questions regarding the material for Thursday's exam.

To the top

Is it possible to determine if a deck is shuffled?

If so, how? If not, why not?

How small can a deck be and still be shuffled?

Good enough for play <--> shuffled <--> random What, precisely, do these words and phrases mean?

The number of ways to write a permutation using disjoint cycle notation can get rather large. How large?

What does it mean for a structure to be a

Tuesday, September 4: We looked at the "birthday problem" and saw that it was very likely that two people would have the same birthday. There was also 99.998% probability that two people would draw the same card.

We looked at the exercises 1.2.1 and had solved 1.2.1.3 for

How many identities are there in

Thursday, September 6: What is the mathematics behind the card trick?

What is the average number of adjacencies in permutations of

We saw that A(s

We also observed that

in S

in S

in S

What sort of patterns exist here?

The

c5:=0;

for pi in s5 do

for j in [1..4] do

if (Abs(j^pi-(j+1)^pi)eq 1) then

c5:=c5+1;

end if;

end for;

end for;

c5;

An explanation of the lines of code:

j^pi- (j+1)^pi looks at the difference between the new positions of j,j+1. If they are adjancent, then the difference will be +1 or -1, so we take the absolute value (Abs) and test to see if that absolute value is equal to 1 (

So far we have two ideas for trying to attack this: 1. Try to come up with a systemic way to count the adjancencies and 2. Use mathematical induction and relate the number of adjacencies in s

We looked at the cycle structre of rho

Some

Thursday, September 13: The number of association schemes for

Is there a systematic way to count these?n1 2 3 4 5 6 7 ...n#schema ? 1 2 5 14 ?? ?? ... ???

Here is the Magma code for running the random commutativity trials in Sn2 3 4 5 6 Prob. 1 1/2 5/24 7/120 11/720

?

ctr:=0;

for j in [1..10000] do

pi:=Random(s10);

for k in [1..10000] do

mu:=random(s10);

if (pi*mu eq mu*pi) then

ctr:=ctr+1;

end if;

end for;

end for;

ctr;

We observed that every 3-cycle may be written as a product of two transpositions.

Ask the same questions for AT(

The number of adjacent transpositions in the factorization of the transposition (a,b) seems to be 2|b-a|-1.

The number of adjacent transpositions in the factorization of the 3-cycle (a,b,c) seems to be 2[|b-a|+|c-b|] -2.

The 4-cycles seem to be trickier.

The factorizations of elements in S

For a permutation

n 2 3 4 5 6 7 8 9 10 ... 17 .... 28 Max order 2 3 4 6 ? ? ? ? >=30 ... >=210 >=2310

d:=matrix(6,6,[ 0,0,0,.5,.5,0, 0,0,0,0,.5,.5, 0,0,0,.5,0,.5, .5,0,.5,0,0,0, .5,.5,0,0,0,0, 0,.5,.5,0,0,0] );

Delta := mtx -> sqrt( sum(sum( (mtx[i,j]-1/6)^2 ,j=1..6), i=1..6) );

plot( [ seq([k, Delta(evalm(d^k))],k=1..10) ] );

The example matrix (

- What is the minimum number of moves required to alphabetize a particular state?
- How would you construct a computer program to alphabetize a particular alphabetizable state?

To the top

- The MAGMA software may be installed on student computers for use in Rose-Hulman course work. It is not to be distributed to others. Your professor will have notified WCC that your class is using MAGMA so that you may install it from the network.
- While connected to the RHIT network connect to the Tibia Software Distribution Service and install the software by following these steps.
- Go to the Start~Run on the Start menu
- Type \\tibia\public\apps in the dialog box and then press Enter
- Double click on the folder called magma2.7
- Double click on the executable magma27.exe
- Make sure that the "install to:" path is c:\ and then click on install.
- You can run MAGMA from the Start~Programs~Magma menu. There are four icons
- Magma - HtmlHelp (help files through the navigation system)
- Magma 2.7 (the program)
- MagmaDocs - Script folder (link to C:\Personal\MagmaDocs a folder for scripts)
- WordPad - Script editor (starts up in C:\Personal)
- When Magma is invoked through the Start Menu it will start up in the C:\Personal\MagmaDocs folder. Thus when users create, edit, and save script files in this folder, the scripts can be easily loaded into Magma without having to supply path names. The installation of Magma has a sample script in C:\Personal\MagmaDocs folder which can be used as a model for other scripts.
- Have fun!

Go to

- the Mathematics Department Home Page.
- my classes page
- my home page
- the top of this page