Discrete Mathematics II, Combinatorics
Math. 325, Sec. 001, Spring 2018
Homework and Announcements
All handouts for the class will be posted as pdf files
after a bullet sign. All these handouts are copyright
material and are intended for the students of this class only.
IN PARTICULAR, DO NOT POST THESE HANDOUTS ON ANY OTHER
WEBSITE OR INTO AN ARCHIVE DATABASE!
For general information regarding this course, please
read the Syllabus here below (to be posted soon), it should answer all
practical questions, like when tests are, what
the grading policies are etc.
Homework and Solutions
Here below the homework (HW) assignments and selected solutions
will be posted. Many of the problems will be exercises
from our textbook. -- Please be sure to read the solutions.
If you have questions or comments feel free to come
and talk to me.
- Homework for Tuesday, January 30. :
We started discussin some preliminaries and listed four
basic principles related to counting:
(C1) Addition Principle, (C2) Product Principle, (C3) The Negation Principle
and (C4) The Division Principle, whose more general form is the
Pigeon Hole Principle (PHP). We discussed some examples on how
this is used to show the existence of certain scenarios. One
corollary of the general form of the PHP is that if n
pigeons are put into n holes and each hole has at most
one pigeon, then each hole must have a pigeon. This can then be used
to prove the Chinese Reminder Theorem.
- Homework for Tuesday, February 6. :
We continued our discussion of the Pigeon Hole Principle (PHP),
and we discussed a particularly nice application of the PHP to
prove a known theorem by Erdős and Szekeres from 1935.
We then discussed a well-know Theorem of Friends and Strangers
that states the in any group of six or more people, where any two
are either friends or strangers, then one can always find a subgroup
of three people that are all mutual friends or all mutual strangers.
Since this is not always possible in a group of five people, then
we know that six is the smallest number of people we must have so
that we can always find three that are all mutually either friends or
strangers. In the language of Ramsey Numbers this means
have R(3,3) = 6, which in the language of graph theory means
that p = 6 is the smallest number such that for any coloring
of the edges of Kp, the complete graph on p
vertices, with two colors (say red and blue), then we can always find a
red K3 or a blue K3 inside
- Homework for Tuesday, February 13. :
We recalled the number of permutations and the
number of r-permtuations of a set containing
n elements. By a circular permutation we mean
an arrangement of elements in a circular fashion where we ignore
the clockwise and counter clockwise rotation of the circle (that is,
a circular arrangement of the elements and any of its rotations is
counted as one.) When counting such circular arrangement it is often
good to fix a designated element, fix that, and then count the
(linear / usual) permutations of the remaining elements. -- By counting
the number of r-permuations in two ways (1) by directly listing
the r elements from a given set of n elements
on one hand, and (2) by first choosing all the r elements
from the given n and then arrange them in a line on the other
hand, we obtain a formula for the number of subsets containing r
elements of a set of n elements. These are the
binomial coefficients and they will be discussed in more
detail later on in the course.
- Homework for Tuesday, February 20. :
By a multiset we mean a collection of elements where
some elements are identical, or in other words, a collection
of elements where we can have more than one copy of each element.
The notions of a submultiset and when two multisets are
equal works in a way analogus to that of usual sets. The number
of permutations of a given multiset is given by a corresponding
multinomial coefficients, a natural generalization
of the binomial coefficients, since when the number k of types is
equal to 2, then the multinomial coefficients reduce
to the binomial coefficients. A typical problem solved by
multinomial coefficient is to count the number of words (real or not real)
one can make from the letters of "MISSISSIPPI". The multinomial coefficients
also count the number of ways we can partition a given (usual) set
into parts, both in a labeled way or in an unlabeled way if all the
parts have the same cardinality. By an r-permutation
of a multiset S we mean an r-string where
no two elements are listed twice (but elements of the same type can.)
Unfortunately, there is no nice formula/expression for the number
of these (unlike when we are dealing with usual sets.) An
r-combination of a multiset S
is simply a sub-multiset of S that contains r
elements. By counting the number of integer solutions to an equation
we obtain a nice binomial formula for the number of
r-combinations of a given multiset. The formula that we
derived to count the number of integer solutions to an integer
equation is interesting in its own right. -- Recall the definition
of a partially ordered set, or a poset for short. It is a binary
relation on a given set which is reflexive, antisymmetric and transitive.
The notion of a subposet is natural, namely it is simply
a subset of the given set of the poset such that any relation that
holds in the subposet also must hold in the given poset. We will discuss
this better next week.