Time: MW 14:30 - 16:25 (2:30 PM - 4:25 PM) Instructors: Travis Scrimshaw (1st half of semester)
Victor Reiner (2nd half of semester)
Location: 2 Vincent Hall Email: reiner@umn.edu
Office hours: Mon and Wed 9:05-9:55am, Tues 1:25-2:15pm, and also by appointment Office: 256 Vincent Hall

## Course Information

This is a course in discrete mathematics, emphasizing both techniques of enumeration (as in Math 5705) as well as graph theory and optimization (as in Math 5707), but with somewhat less depth than in either of Math 5705 or 5707. We plan to cover most of the above text, skipping Chapters 6, 14, 15. We will also supplement the text with some outside material that will be posted on the course webpage.

Math 2243 and either Math 2283 or 3283 (or their equivalent). Students will be expected to know some calculus and linear algebra, as well as having some familiarity with proof techniques, such as mathematical induction.

• Homework: 40%

Homework will be due approximately every other week. The assignments and their due dates will be posted on the course website. All homework is due at the beginning of class. Each problem will be equally weighted.

You are encouraged to work collaboratively on your homework, but you must write up your own solutions, understand your solutions, and indicate your collaborators.

• Midterms: 40%

Two take home midterms.

• Final: 20%

Take-home final exam.

• Final exam - Due: in class Wednesday May 3 or in my mailbox by 4pm.

All exams are open-web/book/note, but no discussion of the exams can occur (e.g., human interaction and online forums).

Late homework will not be accepted. I will excuse late homework/exams for documented emergencies.

## Homework assignments

• Homework 1 - Due: Monday, Feb. 6

Section 1.8: 12, 14, 17, 21, 22, 26, 29, 33

Section 2.5: 1, 3, 4(a), 5, 7, 8

TS1: Define the major index of a permutation $$\sigma = \sigma_1 \cdots \sigma_n$$ by $\operatorname{maj}(\sigma) = \sum_i i,$ where the sum is over all descents, positions $$i$$ such that $$\sigma_i > \sigma_{i+1}$$. Describe how inserting $$n+1$$ before $$\sigma_i$$ changes the major index.

TS2: Show that $\sum_{\sigma \in S_n} q^{\operatorname{inv}(\sigma)} = [n]_q! = [1]_q \cdot [2]_q \cdots [n]_q,$ where $$[i]_q = 1 + q + q^2 + \cdots + q^{i-1}$$ and $$q$$ is a formal parameter.
Note: A bijective proof that replacing inversions with major index (see TS1) results the same polynomial was given by Foata in 1968. The polynomial $$[i]_q$$ (resp. $$[n]_q!$$) is called the $$q$$-integer and (resp. $$q$$-factorial) as it equals $$i$$ (resp. $$n!$$) when $$q=1$$.

• Homework 2 - Due: Monday, Feb. 20

Section 3.8: 8, 9, 12

Section 4.3: 6, 9(b-d), 12, 13, 15

TS3: Consider the recurrence equation $a_n = \alpha a_{n-1} + \beta a_{n-2}$ with $$a_0 = U$$ and $$a_1 = 0$$. Determine both the ordinary and exponential generating functions. Write a closed formula for $$a_n$$.

TS4: Prove the Catalan recurrence formula in terms of Dyck words.

TS5: Determine the (ordinary) generating function for the Catalan numbers.

• Homework 3 - Due: Monday, Mar. 20

Section 7.3: 4, 5, 9, 10,13

Section 8.5: 3, 4, 5, 6, 7, 9, 11

TS6: Let $$L_n$$ and $$U_n$$ denote the number of labeled and unlabeled graphs, respectively, on $$n$$ vertices. Show that $$U_n \geq \frac{L_n}{n!}$$.

TS7: Let $$G$$ be a (not necessarily simple) graph and $$e = \{u,v\} \in E(G)$$. The contraction of $$G$$ along $$e$$, denoted $$G / e$$, is the graph obtained by shrinking $$e$$ to a point. Formally, $$G / e$$ is the graph with vertices ($$V(G) \setminus \{u,v\}) \cup \{uv\}$$ and edges $\{\{a,b\} \mid a,b \notin e\} \cup \{\{a,uv\} \mid \{a, u\} \text{ or } \{b,u\} \in E(G)\}$ and adding loops for every other edge $$\{u,v\}$$ parallel to $$e$$. Let $$\kappa(G)$$ denote the number of spanning trees of $$G$$. Show that $\kappa(G) = \kappa(G \setminus e) + \kappa(G / e).$

• Homework 4 - Due: Monday, Apr. 3

Section 9.2: 3,7

Section 10.4: 5,6,7,11,13(a,b),15

• Homework 5 - Due: Monday, Apr. 24

Section 12.3: 1, 2, 5, 6
Correct the hypotheses of 12.3.6 by assuming also that every vertex has degree 3.
And here is a hint for 12.3.5: first note that the Petersen graph has no cycles shorter than 5-cycles.

Section 13.4 :1, 2, 7, 8, 9(a)

## Topics

We will be covering the following topics:

• Combinatorial objects and techniques
• Sets and subsets
• Sequences
• Permutations
• Inclusion-Exclusion
• Pigeonhole principle
• Estimations
• Algebraic combinatorics
• Binomial coefficients
• Pascal's triangle
• Fibonacci numbers
• Catalan numbers
• Events and probability
• Law of small, large, and very large numbers
• Graph theory
• Definitions
• Eulerian and Hamiltonian walks/cycles
• Trees
• Traveling salesman problem (TSP)
• Graph matchings
• Euler's formula
• Graph colorings
• Geometric combinatorics

As time permits, we will discuss the following topics (not in any particular order):

• Generating functions
• More on Catalan numbers
• More on permutations and their generalizations
• Partitions and tableaux
• Research topics in combinatroics