MATH 2410 Combinatorics I
Students taking this course will be expected to know the basic principles of sets and number systems, linear algebra and analytical geometry. For this reason, MATH 1152 and MATH 1141 are listed as course prerequisites.
This course is divided into the two sections - enumeration and applications. We begin the section on enumeration well-fortified with the methods of proof encountered in the prerequisite courses to study permutations and combinations. The important Pascal numbers are defined from the basis of the binomial theorem for expansion of expressions. By using algebraic and analysis techniques, simple combinatorial identities are established. Next we expand on set algebra theory to formulate the potent principle of inclusion and exclusion. An important application of this principle is seen in the integer solutions of linear equations having unit coefficients where we examine solutions that are bounded above and below. The solution by iteration is discussed. Sufficient conditions for applying the summation method are given in solving a linear homogeneous recurrence of order k. Generating functions are also used to solve non-linear recurrences. The enumeration methods and methods of proof have applications to combinatorial problems as well as areas of applied mathematics dealing with finite sets of objects. Combinatorial probability arises naturally as a result of permutations and combinations. Next we apply the knowledge of generating functions and recurrence relations to study partitions of integers. By including the binomial expansion, we examine random walks in two and three dimensions.