Combinatorics

Introduction to integer partitions: Integer partitions (decompositions of natural numbers into unordered sums of positive integers) are one of my personal favorite objects in mathematics. They are very simple to define and nice to visualize but show up in many deep areas of math, particularly representation theory. This writing is only an introduction to integer partitions and some counting we can do with them. 

Note: at the end when I mention generating functions, I am considering them as formal power series.

Counting, combinatorial classes and generating functions: These notes are a brief introduction to counting in mathematics. We explore combinatorial classes which are sets of objects that have a "size" function defined on them. For any non-negative integer k, the number of objects of size k is finite. Our goal is to determine how many elements there are of size k. Sometimes it's possible to determine these number through direct investigation. However, most of the time this ends up being very difficult. One of the most powerful tools we have in our toolkit is the generating function. Actions on the objects in our combinatorial classes directly correspond to certain algebraic operations on the generating functions.

Catalan numbers count triangulations of convex polygons: The Catalan numbers are one of the most famous sequences in mathematics. They enumerate many different objects: Dyck paths, certain binary trees, permutations which "avoid" any given permutation in S_3, and much more (more than one can count). A well-known example counted by the Catalan numbers are triangulations of convex polygons. A triangulation of a polygon can be thought of as filling in the polygon with triangular pieces like a puzzle. This writing gives a proof that the number of triangulations satisfies the same initial conditions and same recurrence relation as the Catalan numbers. As such, they must be the same sequence. The formula I give for the Catalan numbers is not proved here. However, using the generating function given at the end of the notes, one may derive the formula.

Graduate Combinatorics 1 notes (MTH 880): Combinatorics (or discrete mathematics) is the study of counting and discrete objects. In these notes, the topics covered include: permutations, subsets, integer partitions, set partitions, graphs, the twelvefold way, the principle of inclusion-exclusion, involutions, Lindstrom-Gessel-Viennot lemma (log-concavity and unimodality), generating functions (ordinary and exponential), labeled structures.

Undergraduate Combinatorics 2 notes (MTH 482): The topics covered in these notes include: posets (partially ordered sets), the Möbius function, the Möbius inversion theorem, group actions on finite sets, Burnside's lemma, counting necklaces, Redfield-Polya theory, symmetric functions, the Schur basis, Young tableaux, hooklength formula, Robinson-Schensted-Knuth correspondence, differential posets.