Differential and integral single variable calculus. A bit of linear algebra may be helpful, but is not absolutely required. Office: Vincent Hall Telephone with voice mail : E-mail: reiner math. This is a junior-senior level undergrad course on enumeration, that is, counting things. Topics include elementary counting techniques, including the principle of inclusion-exclusion, along with with more sophisticated techniques, such as generating functions and Polya theory.

Note that Math is class which covers some of this material at a somewhat lower-level, along with some of the graph theory covered in Math Our text will be a sequence of problems to be solved, intended to guide the student through most of the material that is often covered in this class via more traditional lecture methods.

Combinatorics through guided discovery by Kenneth P. In math library, Call no. G44 W55or download it for free. There will be homework assignments due every other week except weeks with exams at the beginning of the Wednesday class, starting with Wednesday February 2. The assignments will be mostly problems from the book, and I will try to hand out brief solutions or solution outlines.

Late homework will not be accepted. I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page with whom they have collaborated. Both the take-home midterm and final exams are to be open-book, open-notes, but there is to be no collaboration ; the only human source you will be allowed to consult is the instructor.

Incompletes will be given only in exceptional circumstances, where the student has completed almost the entire course with a passing grade, but something unexpected happens to prevent completion of the course. Incompletes will never be made up by taking the course again later.

**Lecture 23 . Enumerative Combinatorics (Federico Ardila)**

You must talk to me before the final exam if you think an incomplete may be warranted. This is a 4-credit course, so I would guess that the average student should spend about 8 hours per week outside of class to get a decent grade. Chapter 1, 50, 55, 56, 63 Chapter 1 Supp. Chapter 2 Supp.Mar 21 : Solutions to the final exam are available here.

## PhD Requirements

Note that other solutions are possible. Mar 20 : The final exam can be found here. Uploads to Gradescope will not be accepted after PM. Mar 19 : The first two pages of the final exam can be found here. In particular, you can see the rules and the academic integrity pledge.

Mar 10 : Homework 6 finalized below. No extra problems were added. The late submission deadline on Gradescope for which there is no penalty will be changed to Sunday, March 15th, PM for students wanting extra time. Mar 4 : Homework 6 posted below. More problems to be added later. Feb 26 : Solutions to the second midterm are available here.

Feb 25 : To find your seating assignment, enter your student ID here. To get an idea of where your seat is, consult the map. If you are left-handed, you can ignore your seat assignment and choose any of the left-handed desks. Feb 18 : A practice midterm is available here. Feb 17 : More problems added to Homework 6. Due date changed to Wednesday, February 26th. Feb 14 : Homework 5 posted below.

Feb 5 : More problems added to Homework 4. Feb 3 : Homework 4 posted below.Mathematics is a subject with many facets and many applications.

We offer specializations in actuarial mathematics, computer applications, mathematical biology, and mathematics education. However, many of our math majors do not choose a specialization, and, of course, some students make course choices that prepare them well for graduate study in pure and applied mathematics. We also offer courses in mathematics that apply to industry and courses that could lead to careers in business or finance.

Among their six upper-division MATH courses, CLA students must choose two algebra courses and two analysis courses from the following lists:. See a math faculty adviser well in advance of the senior year to formulate a plan for completing the project. Among their eight upper-division MATH courses, CSE students must choose two algebra courses and two analysis courses from the following lists:.

Note: Students in the Actuarial and Mathematical Biology specializations need only take two semesters of Physics. Each sub-plan below, including "no specialization," represents a specific route toward satisfying the general requirements described above. The minor in Mathematics is offered through CLA. To apply for the Mathematics Minor or ask questions about it, please send an email request to.

Requests are reviewed and approved based on Mathematics course work completed, in progress, and planned. This document addresses only the major requirements; it does not discuss liberal education requirements or the CLA foreign language requirement.

All courses that are in the major or are prerequisite to courses in the major must be passed with a grade of C- or better. Mathematics Degree Programs. Mathematics BS Math :.

Mathematics Minor :. Technical Elective Package. Two courses satisfying the following requirements: At least three credits each Numbered 3xxx or higher Calculus I is a prerequisite Must have a theme or connection, requiring approval by a math faculty adviser. For example, two courses from the same department.

BA Specializations and Sub-Plans. No Specialization - BA.

Actuarial Science Specialization - BA. Computer Applications Specialization - BA. Mathematical Biology Specialization - BA. Mathematics Education Specialization - BA. Three semesters of physics except for the Actuarial and Mathematical Biology Specializations as described above [the first two courses are required prior to admission to the major].

No Specialization - BS.Permission of the instructor. We plan on thoroughly studying Sieve methods, partially ordered sets, and rational generating functions. If a student cannot attend an exam or hand in a homework project on time due to circumstances beyond their control then the instructor may assign appropriate make-up work. Students will not be penalized for absences due to participation in University-approved activities, including athletic or scholastics teams, musical and theatrical performances, and debate activities.

Reasonable accommodation will also be made for students participating in a religious observance. Also, note that grades of Incomplete I are reserved for students who are passing a course but have not completed all the required work because of exceptional circumstances. The student must show exceptional circumstances why requirements cannot be met.

A request for an incomplete grade has to be made in writing with supporting documentation, where appropriate. Students at Florida Atlantic University are expected to maintain the highest ethical standards. Academic dishonesty, including cheating and plagiarism, is considered a serious breach of these ethical standards, because it interferes with the University mission to provide a high quality education in which no student enjoys an unfair advantage over any other.

Academic dishonesty is also destructive of the University community, which is grounded in a system of mutual trust and places high value on personal integrity and individual responsibility. Harsh penalties are associated with academic dishonesty. For more information, see University Regulation 4. Sep 9 —Sep V-variations and unimodal sequences Involutions Determinats. Nov 4 — Nov 6 Binomial pose ts and generating functions, an application to permutation enumeration.

Course Syllabi. Weekly outline. Cumulative Performance.Permission of the instructor. Introduction to enumeration. Sets and multisetspermutations, sieve methods, partially ordered sets, lattices, incidence algebra, Moebius inversion, and generating functions. If a student cannot attend an exam or hand in a homework project on time due to circumstances beyond their control then the instructor may assign appropriate make-up work.

Students will not be penalized for absences due to participation in University-approved activities, including athletic or scholastics teams, musical and theatrical performances, and debate activities. Reasonable accommodation will also be made for students participating in a religious observance.

Also, note that grades of Incomplete I are reserved for students who are passing a course but have not completed all the required work because of exceptional circumstances. The student must show exceptional circumstances why requirements cannot be met. A request for an incomplete grade has to be made in writing with supporting documentation, where appropriate. Students at Florida Atlantic University are expected to maintain the highest ethical standards.

Academic dishonesty, including cheating and plagiarism, is considered a serious breach of these ethical standards, because it interferes with the University mission to provide a high quality education in which no student enjoys an unfair advantage over any other. Academic dishonesty is also destructive of the University community, which is grounded in a system of mutual trust and places high value on personal integrity and individual responsibility.

Harsh penalties are associated with academic dishonesty. For more information, see University Regulation 4. Course Syllabi. Weekly outline. Cumulative Performance.There are no official prerequisites for this course, though familiarity with combinatorics is assumed. Stanley, R. Enumerative Combinatorics. I and II. ISBN: hardback: vol. I ; paperback: vol.

I ; hardback: vol. ISBN: Extremal Graph Theory. New York, NY: Dover, Jukna, S. Extremal Combinatorics. There are eight problem sets, each weighted equally for your grade.

Collaboration is encouraged with a few simple rules. On every problem not more than four people can collaborate. For each problem, all collaborators should be listed. Don't show me this again. This is one of over 2, courses on OCW.

Find materials for this course in the pages linked along the left. No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates. Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW. Made for sharing.

Download files for later. Send to friends and colleagues. Modify, remix, and reuse just remember to cite OCW as the source. Main Textbook There are four main textbooks used for this class: Stanley, R. Problem Sets There are eight problem sets, each weighted equally for your grade. Exams There are no exams in this course.

## Enumerative Combinatorics

Grading The entire grade is based on the eight problem sets. Need help getting started? Don't show me this again Welcome!An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set.

The precise requirements for an enumeration for example, whether the set must be finiteor whether the list is allowed to contain repetitions depend on the discipline of study and the context of a given problem. Some sets can be enumerated by means of a natural ordering such as 1, 2, 3, 4, In some contexts, such as enumerative combinatoricsthe term enumeration is used more in the sense of counting — with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements.

In combinatorics, enumeration means countingi. There are flourishing subareas in many branches of mathematics concerned with enumerating in this sense objects of special kinds. For instance, in partition enumeration and graph enumeration the objective is to count partitions or graphs that meet certain conditions.

In set theorythe notion of enumeration has a broader sense, and does not require the set being enumerated to be finite.

When an enumeration is used in an ordered list context, we impose some sort of ordering structure requirement on the index set. While we can make the requirements on the ordering quite lax in order to allow for great generality, the most natural and common prerequisite is that the index set be well-ordered.

According to this characterization, an ordered enumeration is defined to be a surjection an onto relationship with a well-ordered domain. This definition is natural in the sense that a given well-ordering on the index set provides a unique way to list the next element given a partial enumeration.

The most common use of enumeration in set theory occurs in the context where infinite sets are separated into those that are countable and those that are not.

This definition can also be stated as follows:. We may also define it differently when working with finite sets. In this case an enumeration may be defined as follows:. In the first definition it varies whether the mapping is also required to be injective i. In some applications especially those concerned with computability of the set Sthese differences are of little importance, because one is concerned only with the mere existence of some enumeration, and an enumeration according to a liberal definition will generally imply that enumerations satisfying stricter requirements also exist.

Enumeration of finite sets obviously requires that either non-injectivity or partiality is accepted, and in contexts where finite sets may appear one or both of these are inevitably present. The following table gives the first few values of this enumeration:. In set theorythere is a more general notion of an enumeration than the characterization requiring the domain of the listing function to be an initial segment of the Natural numbers where the domain of the enumerating function can assume any ordinal.

This more generalized version extends the aforementioned definition to encompass transfinite listings. More generally, it is a theorem of ZF that any well-ordered set can be enumerated under this characterization so that it coincides up to relabeling with the generalized listing enumeration.

If one also assumes the Axiom of Choicethen all sets can be enumerated so that it coincides up to relabeling with the most general form of enumerations. Indeed, in Jech's book, which is a common reference for set theorists, an enumeration is defined to be exactly this. Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.

### MA241 Combinatorics

Formally, the most inclusive definition of an enumeration of a set S is any surjection from an arbitrary index set I onto S.

In this broad context, every set S can be trivially enumerated by the identity function from S onto itself. If one does not assume the axiom of choice or one of its variants, S need not have any well-ordering. Even if one does assume the axiom of choice, S need not have any natural well-ordering. This general definition therefore lends itself to a counting notion where we are interested in "how many" rather than "in what order.

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