Scores on the Final: 200(1), 190s(6), 180s(3), 170s(7), 160s(1), 150s(3), 140s(2), 130s(2), 110s(1), absent(2).
Grades in the Course: A+(3), A(9), A-(4), B(1), B-(3), C+(4), C-(2), F(2). Note: for the first time since 1978, a student got an A by fiat. One student had B+ on points, but 194 on the final, and so got the A.
Visitation in my office Tuesday, May 18, Noon-12:30 or by appointment. I should be around "most" of the summer.
I hope you have an enjoyable and productive summer. -- BR
F 1/23 -- Second day of class. Distribution of Problem-solving
handout.. A slide show of
coloring checkerboards or broken checkerboards with dominos or other
shapes. Additional review of sections 1.1 through 1.3. A further look
at the coloring, leading to a proof that there are always at least two
monochromatic triangles.
M 1/26 -- The first Bonus Notes, covering the definition of complete
graphs, and giving a written version of the proof of the proof from
1/23. (See 2.4 -- 313.) We discussed Latin squares and orthogonal
Latin squares with a bunch of examples and a sense of doing
crossword puzzles and moved on to the Pigeonhole Principle (PHP).
W 1/28 -- The second Bonus Notes, covering the definition of Latin
squares and some constructions discussed in class on 1/26. The PHP
in some contrived settings, and some not so much (such as the Chinese
Remainder Theorem.)
Prof. Muncaster brought down the painting of 10 by 10 orthogonal Latin
squares that hangs in his office, 274 Altgeld.
F 1/30 -- The third Bonus Notes, covering Dirichlet's Theorem about
multiples of irrational numbers. The first assignment, Homework
One, was distributed. [Note, this has the corrected version
of 9a.] In class, we also did the statement of the general PHP,
Dirichlet's Theorem and the existence of monotone subsequences of
length n + 1 in every sequence of length n2 + 1.
M 2/2 -- Bonanza of handouts! Bonus Notes 0, covering the first day
of class, then Bonus Notes 4a (On putting 3 points in a square) and
Bonus Notes 4b (on how 10 orange edges on K6 forces an
orange triangle), as well as the corrected homework 1. Discussion
of these, plus a slight variation of Polya-Szekeres to mn+1 numbers
and sequences of m+1 non-increasing or n+1 non-decreasing.
W 2/4 -- Moving on to the more "normal" part of the course, we look
at 3.1 and 3.2 and some of basics of counting. Handout -- the phone
and email list. Most of what was done is closely related to the text.
F 2/6 -- The second assignment, Homework
Two, was distributed, as were handwritten solutions to Homework
One. About half the class consisted of these solutions, gone over
carefully. The rest was a continuation of 3.2 and 3.3, finishing up
with C(n,r) and its combinatorial interpretations.
M 2/9 -- More combinatorial examples from Chapter 3.
W 2/11 -- Homework 1 returned, along with a supplemental page of
comments. We continued through section 3, and nearly finished it.
F 2/13 -- Your aching prof managed to make it through the solution
sheet to the second homework (all four pages).
Nothing broken, just a few bruises and strains. That will
teach me to pay attention where I walk.
The third assignment, Homework
Three, was distributed.
Sorry for the delay and brevity in writing up this last week's
classes. I'll put in more detail later if anyone asks.
M 2/16 --A general remark: every math,
science or engineering major winds up being a teacher either formally
(e.g. high school, graduate TA, prof) or informally (e.g. explaining
some new software to your group at work.) I have written a TA training
guide used here and at many other schools at one time or another
(well, it's free), and you can find it here:
"Chalking It Up"
. You'll be able to judge for yourself how well I meet my own
guidelines. The informal midterm review was distributed, to be
collected later. Bonus notes 5, on playing cards. A start on
combinatorial identities, with an overly-enthusiastic introduction to
generating functions.
W 2/18 -- Homework 2 returned and discussed. The supplemental comments
took only half a page, so the rest of the handout was Bonus notes 6, on
generating functions. More versions of the proof of the binomial theorem.
Questions on homework 3.
In problem 8, do you mean "a particular red suit" or that four of the cards
are red (possibly a mixture of hearts and diamonds)?
A. I mean that if you look at the color of the cards, four of them
will be red and one will be black.
Q. What's going on with problem 9?
A. These involve identities with, eg kC(n,k), that can be found in
the section or in class discussion. Note added after 2/20 class: Curse
you, Brualdi, for not having the Vandermonde Identities in the body of
the chapter, but only as an exercise! If such a thing should happen
again, please ask sooner than Thursday night. There are also other
combinatorics books out there.
F 2/20 -- Most of the class was devoted to talking about Homework 3,
whose solution was distributed. Review Questions
for the exam were
distributed. Note that the exam will cover chapters 1,2,3 and not
combinatorial identities.
M 2/23 -- More on combinatorial identities. Bonus Notes 7. a summary
of Chapter 3, was distributed, along with Bonus Notes 8, a two-sider
on combinatorial identities. The fourth assignment, Homework
Four, was distributed. It is due Friday 3/5.
W 2/25 -- Homework 3 was returned, supplemental comments distributed
and discussed.
Mostly review for the exam and a discussion of the review
questions previously distributed.
F 2/27 -- The first exam. I will not post it. I also don't write
solutions for the tests.
M 3/1 -- The first exam, returned and discussed at great length.
We ended class with the first part of a discussion of chains, antichains
(what the book calls clutters) and Sperner's Theorem.
W 3/3 -- Sperner's Theorem discussed at great detail, since it
is more sophisticated than anything else in the course so far.
The multinomial theorem is stated and proved as well. HW 4 is
delayed until Monday 3/8. An oddly relevant discussion of religious
restrictions in 19th century British universities.
F 3/5 -- Chapter 5 finishes with a discussion of the Binomial
Theorem in full generality; (1+x)t, where t is *not*
a positive integer, and convergence is only assured for |x| < 1. Special
emphasis on when t is a negative integer and on t = -1/2. Chapter 6
begins with the Principle of Inclusion and Inclusion, and examples
based on letter patterns in a word. Homeworks, until the Break at least,
will be due on Mondays, not on Fridays.
M 3/8 -- We discuss homework four, and I pass out Homework
Five, even if we don't write the webpage up until Wed. afternoon.
A waltz through chapter 6. Maria Boca pointed out a subtlety on
problem 9. When I wrote it,
I had in mind the original definition of a binomial coefficient,
in which "-1 choose 7", say, would be automatically 0. Of course,
we have a new second definition of binomial coefficient, in which
this expression is not zero. So... make the sum go from 0 to n,
please. And...the first Bonus Notes 9, on combinatorial identities.
W 3/10 -- More on inclusion and exclusion, and problems that might
better be done using generating functions. Derangements of various
kinds. Something about rook polynomials. A corrected version of the
ROMANTIC CORN TACO problem is discussed.
F 3/12 -- Homework 4 retrospective.
The general version of the principle of inclusion
and a few applications. Handouts are promised. They're coming on
Friday, quite a few of them in fact. A hint about recurrences and how
they can be used.
M 3/15 -- Someone needed a vacation. I gave out Handout 9 (on the
General Principle of Inclusion and Exclusion), but: identified it
with Math 213 incorrectly, and neglected to notice that this was
the second Bonus Handout 9. (See 3/8.) Bonus Handout 10 on Rook
Polynomials was, relatively speaking, numbered correctly. However, it
was dated 3/25, not 3/15. Homework 5 solutions were passed out, and
Homework
Six was distributed. Contrary to the statement on the sheet, it's
actually due W 3/31. We basically decided that the second test would
be on F 4/9.
W 3/17 -- Full bore into chapter 7, constant-coefficient linear
recurrences, and some of the amazing properties of the Fibonacci
numbers.
F 3/19 -- More fun with the Fibonacci numbers and linear recurrences,
plus stories about Richard Feynman. Those who were there had a good time.
Spring Break
M 3/29 -- Welcome back for the second half. We talked more about
linear recurrences, how to solve them, and how sometimes the best
way to treat a cold is to turn it into a pneumonia. Homework
Seven was distributed.
W 3/31 -- Homework Six was collected, solutions to Homework Six were
distributed and most of the class period was devoted to discussing a
few of them. We talked at the end about sequences with forbidden
subpatterns.
F 4/2 -- More on sequences with forbidden subpatterns. A discussion of
how the generating function method allows one to derive the general
solution to linear recurrence equations. A handout on the method of
partial fractions (see your "techniques of integration" section from
calculus). It can be worthwhile to come to class!
M 4/5 -- HW 7 due and discussed. A few other examples presented.
Review questions discussed
W 4/7 -- Review for test 2.
F 4/9 -- Test 2.
M 4/12 -- We move on. Homework
Eight was distributed. Test 2 was discussed. Key statistic:
for the 60% of the class who's been attending regularly, the average
was the same as for Test 1; for the other 40%, the average was 15
points lower. We began to finish chapter 7 and discussed the
"geometric" example of 7.6, which serves to introduce the Catalan numbers.
W 4/14 -- We finish the discussion of Catalan numbers (see also
Chapter 8), and the connection to parenthesizing n symbols with a
non-associative binary operation. Some discussion of exponential
generating functions. Supplemental notes on generating functions.
F 4/16 -- More on exponential generating functions, and some talk
about homework problems. Some relevant hints: try to factor
denominators in generating functions as (1 - r1 x)(1 - r2 x) ... in
order to use the partial fraction method most efficiently.
Supplemental notes on exponential generating functions.
M 4/19 -- Homework 8 collected and solutions distributed. Also, Homework
Nine was distributed. More bonus notes on generating functions.
We discuss the number of ways of decomposing a 3 x 2n board into
1 x 2 dominos. This involves an auxiliary sequence.
W 4/21 -- Handout called "Section 8.2 Untangled, Part 1". We talk
about polynomials, the difference operaton,
difference tables, Stirling Numbers of the
second kind and how it becomes trivial to sum polynomials from 0 to n.
F 4/23 -- Another handout called "Section 8.2 Untangled, Part 2",
This is discussed, along with other topics from 8.2, including the
combinatorial meaning of Stirling Numbers of the Second Kind. For the
benefit of those doing homework 9, we also briefly discuss
Stirling Numbers of the First Kind and conjugate partitions.
M 4/26 -- Homework 9 due and discussed.
W 4/28 -- Homeworks 8 and 9 returned. Review for the exam.
F 4/30 -- Third hour test
M 5/3 -- Third hour test to returned. Fun with partitions.
W 5/5 -- ICES forms. Fun with generating functions, Bell Numbers
and Stirling Numbers of the Second Kind.
Th 5/13 -- Final Examination, 8-11 am.