Discrete Mathematics II, Combinatorics

Math. 325, Sec. 001, Spring 2018

Homework and Announcements

Urgent stuff

Please NOTE! 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!

General Info

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.

  1. 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.
  2. 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 our Kp.
  3. 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.
  4. 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.

Exam Info