Krishna Menon: Two Topics in Enumerative Combinatorics

Abstract: The first topic we will explore is Hyperplane Arrangements. A hyperplane arrangement is a finite collection of affine hyperplanes in Euclidean space. Our goal is to interpret the coefficients of a certain
polynomial associated to an arrangement called its characteristic polynomial. This is done by defining an appropriate statistic on the regions into which the arrangement breaks the space. We will be focusing
on a popular class of arrangements called 'Deformations of Reflection Arrangements'.

The second topic is Pattern Containment. Patterns are usually studied for permutations, where the main problem is counting permutations that do not contain particular types of subsequences. We will however be
focusing on patterns in binary words. The numbers we study are of the form B(n, k, p), which is the number of binary words of length n that have exactly k occurrences of the subsequence (pattern) p. After looking
at some results about these numbers, we end with a conjecture and directions for future research.