Acknowledgements

I would like to thank each and everyone who attended the UDGRP and gave presentations. A shoutout to Arkapriyo Hore, for helping me out whenever needed, and also, Anasmit Pal for helping me out at times, when things were beyond the scope of my knowledge.

Abstract

Combinatorics is a branch of mathematics focused on the study of discrete objects and their arrangements. It deals with counting, enumerating, and analyzing configurations of objects that can be distinct or identical, finite or infinite, and arranged in various ways. Over the years as this field of mathematics has matured it has become exceedingly harder to provide purely combinatorial arguments for the questions that continue to arise. Thus, there has been a need to introduce the interplay of combinatorics with various methods ranging from algebraic to analytic to probabilistic. In this UDGRP, we will mainly be focusing on the algebraic and geometric methods, and at times, may call upon some probabilistic methods, if necessary.

Main References

• Handbook of Enumerative Combinatorics
• A Course in Enumeration – Martin Aigner
• Bijective Combinatorics – Nicholas Loehr
• Enumerative Combinatorics Vol. 1 & 2 – Richard P. Stanley
• A Course in Combinatorics – J.H. van Lint & R.M. Wilson

Overview

Day 1: Brief introduction to Group Theory and Generating Functions

This was the very first session of the Enumerative Combinatorics UDGRP. We used the time to get introduced to each other as well as everyone’s motivation.

• Scribbles: Available here
• Topics Covered:
  • Generating Functions
  • Basic Group Theory
  • Asymptotics
  • A brief motivation for Algebraic and Probabilistic Combinatorics
• Date: 28th November, 2024
• References:
  • Visual Group Theory – Nathan Carter
  • Abstract Algebra – Dummit and Foote
  • Generating Functionology – Herbert S. Wilf

Day 2: An insight into Geometric and Topological Combinatorics

This was the second session of the Enumerative Combinatorics UDGRP. We used the time to introduce some areas in geometric and topological combinatorics, and also provide a brief introduction to additive combinatorics. For further reading, look into the books in the reference.

• Scribbles: Available here
• Topics Covered:
  • Proof of Cauchy-Davenport Theorem
  • Basic Idea of Planarity
  • Introduction to Polytopes
  • Introduction to Hyperplane Arrangements
• Date: 1st December, 2024
• References:
  • The Cauchy Davenport Theorem
  • Lecture Notes in Hyperplane Arrangements
  • Lectures in Polytopes – Gunter M. Ziegler

Day 3: Asymmetric Colouring and the EGZ Theorem

• Presentation: Asymmetric Colouring – Daibik Barik
• Notes of Presentation: Available here
• Scribbles: Available here
• Topics Covered:
  • Chevalley-Warning Theorem
  • Proof of Erdos, Ginzburg, Ziv Theorem using counting argument
• Date: 5th December, 2024
• References:
  • Zero Sum Sets of Prescribed Size

Day 4: Planarity and Kuratowski’s Theorem

• Presentation: Kuratowski’s Theorem – Aayusman Mallick & Arkaprovo Das
• Notes of Presentation: Available here
• Date: 8th December, 2024

Day 5: Lovasz Local Lemma

• Presentation: Lovasz Local Lemma – Adrija Chatterjee
• Notes of Presentation: Available here
• Date: 15th December, 2024

Day 6: Partition Function and Euler’s Pentagonal Number Theorem

• Presentation: Euler’s Pentagonal Number Theorem – Anuvab De & Priyankar Biswas
• Notes of Presentation: Available here
• Date: 19th December, 2024

Day 7: A dive into Ramsey Numbers and Random Graphs

• Presentation: Counting Probabilities – Subhojit Maji
• Notes of Presentation: Available here
• Date: 22nd December, 2024

Day 8: Guest Session on some ideas in Topological Combinatorics

• Guest Speaker: Anasmit Pal
• Session Highlights:
  • Sperner’s Lemma
  • Brouwer’s Fixed Point Theorem
  • Fair Division Problem
• Session Notes: Available here
• Date: 29th December, 2024