Percy Alexander MacMahon: Collected Papers

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Title Contents Introduction Preface Bibliography of MacMahon 1. Symmetric Functions (Part 1) 1.1 Introduction and Commentary 1.2 The Combinatorial Development of Symmetric Functions 1.3 References 1.4 Summaries of the Papers [7] Note on an algebraic identity [11] On Symmetric Functions, and in particular on certain Inverse Operators ... [12] Note on the development of an algebraic fraction [16] A new theorem in symmetric functions [17] Note on rationalisation [25] The law of symmetry and other theorems in symmetric functions 2. Symmetric Functions (Part 2) 2.1 Introduction and Commentary 2.2 The Algebra of Partitions, by Philip Hall 2.3 References 2.4 Summaries of the papers [10] Algebraic identities arising out of an extension of Waring’s formula [15] The multiplication of symmetric functions [28] Properties of a complete table of symmetric functions [31] Memoir on a new theory of symmetric functions [32] Symmetric functions and the theory of distributions [35] Second memoir on a new theory of symmetric functions [39] Third memoir on a new theory of symmetric functions [43] Fourth memoir on a new theory of symmetric functions [88] Small contribution to combinatory analysis [101] The algebra of symmetric functions [102] An American tournament treated by the calculus of symmetric functions [103] Chess tournaments and the like treated by the calculus of symmetric functions 3. The Master Theorem 3.1 Introduction and Commentary 3.2 The Master Theorem 3.3 Good’s Proof of Dyson’s Conjecture 3.4 References 3.5 Summaries of the Papers [45] A certain class of generating functions in the theory of numbers [61] The sums of powers of the binomial coefficients 4. Permutations 4.1 Introduction and Commentary 4.2 A Factorization Theorem and its Consequences 4.3 References 4.4 Summaries of the Papers [41] Applications of a theory of permutations in circular procession to the theory of numbers [48] Self-conjugate permutations [53] Solution du problème de partition d’ou résulte le dénombrement des genres distincts d’abaque relatifs aux équations à n variables [81] The problem of ‘derangement’ in the theory of permutations [82] The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects [83] The superior and inferior indices of permutations [86] Two applications of general theorems in combinatory analysis: (1) to the theory of inversions of permutations; (2) to the ascertainment of the numbers of terms in the development of a determinant which has amongst its elements an arbitrary number of zeros [95] Permutations, lattice permutations, and the hypergeometric series [108] Prime lattice permutations 5. Compositions and Simon Newcomb’s Problem 5.1 Introduction and Commentary 5.2 The Dillon-Roselle Solution of Simon Newcomb’s Problem 5.3 C. Long’s Conjecture 5.4 References 5.5 Summaries of the Papers [40] Yoke-chains and multipartite compositions in connexion with the analytical forms called “trees” [42] The combinations of resistances [46] Memoir on the theory of the compositions of numbers [71] Second memoir on the compositions of numbers 6. Perfect Partitions 6.1 Introduction and Commentary 6.2 Complementing Sets of Integers 6.3 References 6.4 Summaries of the Papers [20] Certain special partitions of numbers [38] The theory of perfect partitions of numbers and the compositions of multipartite numbers [107] The partitions of infinity with some arithmetic and algebraic consequences [109] The prime numbers of measurement on a scale 7. Distributions upon a Chess Board and Latin Squares 7.1 Introduction and Commentary 7.2 Redfield’s Notation 7.3 The Theory of Group-Reduced Distributions by J. H. Redfield 7.4 References 7.5 Summaries of the Papers [52] A new method in combinatory analysis with application to Latin squares and associated questions [57] Combinatory analysis. The foundations of a new theory [89] Combinations derived from m identical sets of n different letters and their connexion with general magic squares 8. Multipartite Numbers 8.1 Introduction and Commentary 8.2 Multipartite Partitions and Gordon’s Theorem 8.3 References 8.4 Summaries of the Papers [79] On compound denumeration [87] Seventh memoir on the partition of numbers. A detailed study of the enumeration of the partitions of multipartite numbers [111] Dirichlet series and the theory of partitions [114] The enumeration of the partitions of multipartite numbers [117] Euler’s phi-function and its connexion with multipartite numbers 9. Partitions 9.1 Introduction and Commentary 9.2 Generating Functions 9.3 Partition Asymptotics 9.4 Congruence Properties of Partition Functions 9.5 Identities for Partition Functions 9.6 References 9.7 Summaries of the Papers [51] Memoir on the theory of the partition of numbers -- Part I [94] On the partitions into unequal and into uneven parts [97] Note on the parity of the number which enumerates the partitions of a number [110] The theory of modular partitions [118] The parity of p(n), the number of partitions of n, when n = mu y [65] Note on the Diophantine inequality lambda x >= mu y [66] On a deficient multinomial expansion [67] Memoir on the theory of the partitions of numbers -- Part III [68] The Diophantine equation x^n - N y^n = z [73] Memoir on the theory of the partitions of numbers -- Part IV. On the probability that the successful candidate at an election ballot may never at any time have fewer votes than the one who is unsuccessful; on a generalization of this question; and on its connexion with other questions of ... 11. Plane Partitions (Part 1) 11.1 Introduction and Commentary 11.2 The Generating Function for Plane Partitions 11.3 References 11.4 Summaries of the Papers [77] Memoir on the theory of the partitions of numbers -- Part V [105] The connexion between the sum of the squares of the divisors and the number of the partitions of a given number 12. Plane Partitions (Part 2) and Solid Partitions 12.1 Introduction and Commentary 12.2 An Introduction to the Knuth-Bender Theorems on Plane Partitions 12.3 References 12.4 Summaries of the Papers [56] Partitions of numbers whose graphs possess symmetry [78] Memoir on the theory of the partitions of numbers -- Part VI. Partitions in two-dimensional space, to which is added an adumbration of the theory of the partitions in three-dimensional space Contents of Volume II