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Public viva-voce Notification Date: Tuesday, 21 January 2025 Time: 11:00 AM Venue: Seminar Hall Two Topics in Enumerative Combinatorics Krishna Menon P Chennai Mathematical Institute. 21-01-25 Abstract The basic problem of Enumerative Combinatorics is that of counting the number of elements of a finite set. As one might guess, this covers a wide range of mathematical topics and addresses problems in various areas. We will be dealing with two topics in Enumerative Combinatorics: Hyperplane Arrangements and Pattern Avoidance. Both topics have several enumerative questions which have received significant attention over the past few decades. First, we explore hyperplane arrangements. A hyperplane arrangement is a finite collection of affine hyperplanes in Euclidean space. Our goal is to bijectively count the regions into which an arrangement divides the space. We will be focusing on a popular class of arrangements called 'Deformations of Reflection Arrangements'. Next, we move on to pattern avoidance. Pattern avoidance is usually studied for permutations, where the main problem is counting permutations that do not contain particular types of subsequences. We will be focusing on pattern avoidance in circular permutations -- permutations where numbers are arranged on a circle instead of a line. We obtain our results by relating pattern-avoiding circular permutations to well-studied objects such as binary words. All are invited to attend the viva-voce examination.
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