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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.18 Enumerative and Algebraic Combinatorics

Doron Zeilberger

1 Introduction

Enumeration, otherwise known as counting, is the oldest mathematical subject, while algebraic combinatorics is one of the youngest. Some cynics claim that algebraic combinatorics is not really a new subject but just a new name given to enumerative combinatorics in order to enhance its (former) poor image, but algebraic combinatorics is in fact the synthesis of two opposing trends: abstraction of the concrete and concretization of the abstract. The former trend dominated the first half of the twentieth century, starting with Hilbert’s “theological” proof of the fundamental theorem of invariants, in which he showed by abstract means that certain invariants existed, ...

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