An Introduction to Algebraic Combinatorics

TL;DR

This introductory paper on algebraic combinatorics covers formal power series, integer partitions, permutations, subtractive methods, and symmetric polynomials, providing foundational concepts and proofs suitable for graduate students.

Contribution

It offers a comprehensive, rigorous introduction to key topics in algebraic combinatorics, including proofs of classical identities and methods, tailored for a graduate course.

Findings

Proves Jacobi's triple product identity

Introduces Bender--Knuth involutions for Littlewood--Richardson rule

Provides extensive exercises for learning

Abstract

This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving Jacobi's triple product identity), permutations (Lehmer codes, cycles) and subtractive methods (alternating sums, cancellations and inclusion-exclusion principles, with a particular focus on sign-reversing involutions and determinants). The last chapter introduces symmetric polynomials and proves the Littlewood--Richardson rule using Bender--Knuth involutions (a la Stembridge). The appendix contains over 200 exercises (without solutions).