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Combinatory Analysis, Volumes I and II
Percy A. MacMahon

AMS Chelsea Publishing
1960; 642 pp; hardcover
Volume: 137
Reprint/Revision History:
third printing with corrections 1984; first AMS printing 2001
ISBN-10: 0-8218-2832-0
ISBN-13: 978-0-8218-2832-8
List Price: US$50
Member Price: US$45
Order Code: CHEL/137.H
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By "combinatory analysis", the author understands the part of combinatorics now known as "algebraic combinatorics". In this book, the classical results of the outstanding 19th century school of British mathematicians are presented with great clarity and completeness.

From the Introduction (1915): "The object of this work is, in the main, to present to mathematicians an account of theorems in combinatory analysis which are of a perfectly general character, and to show the connection between them by as far as possible bringing them together as parts of a general doctrine. It may appeal also to others whose reading has not been very extensive. They may not improbably find here some new points of view and suggestions which may prompt them to original investigation in a fascinating subject ...

"In the present volume there appears a certain amount of original matter which has not before been published. It involves the author's preliminary researches in combinatory theory which have been carried out during the last thirty years. For the most part it is original work which, however, owes much to valuable papers by Cayley, Sylvester, and Hammond."


Graduate students and research mathematicians.

Table of Contents

Section I. Symmetric Functions
  • Elementary theory
  • Connexion with the theory of distributions
  • The distribution into parcels and groups in general
  • The operators of the theory of distributions
  • Applications of the operators \(d\) and \(D\)
Section II. Generalization of the Theory of Section I
  • The theory of separations
  • Generalization of Waring's formula
  • The differential operators of the theory of separations
  • A calculus of binomial coefficients
  • The theory of three identities
Section III. Permutations
  • The enumeration of permutations
  • The theory of permutations
  • The theory of displacements
  • Other applications of the master theorem
  • Lattice permutations
  • The indices of permutations
Section IV. Theory of the Compositions of Numbers
  • Unipartite numbers
  • Multipartite numbers
  • The graphical representation of the compositions of tripartite and multipartite numbers
  • Simon Newcomb's problem
  • Generalization of the foregoing theory
Section V. Distributions Upon a Chess Board, to Which is Prefixed a Chapter on Perfect Partitions
  • Theory of the perfect partitions of numbers
  • Arrangements upon a chess board
  • The theory of the latin square
Section VI. The Enumeration of the Partitions of Multipartite Numbers
  • Bipartite numbers
  • Tripartite and other multipartite numbers
  • Tables
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