# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 146: Algebraic Combinatorics

Approved: 2006-02-28, A. Schilling
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Herbert Wilf, "Generatingfunctionology", 2005, \$39
http://www.math.upenn.edu/%7Ewilf/DownldGF.html
Search by ISBN on Amazon: 1-56881-279-5
Prerequisites:
((MAT 022A or MAT 027A or BIS 027A, MAT 108) or MAT 067)); MAT 145.
Suggested Schedule:
 Lecture(s) Sections Comments/Topics 2 Chapter 1 Recurrences, generating functions 3 Chapter 1 Examples: Fibonacci numbers, binomial coefficients, Stirling numbers of second kind 1 Chapter 1 Bell numbers, exponential generating functions 6 Chapter 2 Formal power series, calculus of formal power series. Examples: Fibonacci numbers (again), Catalan numbers, derangements, Moebius function 3 Chapter 3 Exponential formula: Cards, decks, hands 5 Chapter 3 Examples: Permutations and their cycles, involutions, 2-regular graphs, connected bipartite graphs, labeled trees, money changing problem, partitions of integers, rooted trees and forests 3 Chapter 4 Sieve method, fixed points in permutations, k-cycles, Stirling numbers (again), rooks on chessboard, snake oil method 5 Chapter 4; Biggs Chapter 27 Cycle index of symmetric group, cyclic, dihedral symmetry symmetries in 3 dimensions, Polya theory
Learning Goals:
AS A CAPSTONE: Using advanced algebraic tools in MAT 146, students will develop and deepen their combinatorial thinking that they were introduced to in MAT 145. Via this subject area expertise, they will gain mastery in this area of specialization and improve their ability to communicate mathematics at a capstone level, commensurate with that expected of one with an undergraduate degree in mathematics.