Index:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu

An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu

Title An introduction to Combinatory analysis
Author Percy Alexander MacMahon
Year 1920
Publisher Cambridge [Eng.] The University press
Source djvu
Progress To be proofread
Transclusion Index not transcluded or unreviewed
OCLC 1047475007

Table of Contents


Chapter I

Elementary Theory of Symmetric Functions

art. page
1—3. Definitions. The Partition Notation. The Power-Sums 1
4—5. The Elementary Function. Homogeneous Product-Sums 4
6—8. Relations between the important series of functions 5
9—10. Combination and Permutation of letters. Partitions and Compositions of numbers 8
11—13. Order of arrangement of combinations, permutations, partitions and compositions. Dictionary or Alphabetical Order 8


Chapter II

Opening of the Theory of Distributions

14—15. Definite way of performing algebraical multiplication 11
16—20. Distribution of letters or objects into boxes. Specifications of objects and boxes. Multinomial Theorem. Distribution Function 12
21—23. Examples of Distribution. Dual interpretation of Binomial Theorem 15
24—27. Interpretation of the product of two or more monomial symmetric functions 17
28—29. The multiplication of symmetric functions. Derivation of formulæ. The symbol of operation 22
30—31. Operation of upon a product of functions. Connexion with the compositions of 25


Chapter III

Distribution into different boxes

32—33. Determination of the enumerating function in the case of two boxes 27
34—37. The general theory of any number of boxes. Operation of upon products of product-sums. Numerical methods and formulæ 29
38—39. Restriction upon the number of similar objects that may be placed in similar boxes. Operation of in this case 33

Chapter IV

Distribution when Objects and Boxes are equal in number

40—42. Solution by means of product-sums. Interchange of Specification of Objects and Boxes. Theorem of symmetry in the algebra of product-sums. Employment of the symbol 36
43—47. Pairing of objects of two different sets of objects. Specification of a distribution. Restriction upon the number of similar objects that can be placed in similar boxes. The operation of 38
48—49. Enumeration of rectangular diagrams involving compositions of numbers 42
50—51. Equivalences of certain distributions 44


Chapter V

Distributions of given specification


52—58. New functions which put the specification of a distribution in evidence. Proof of symmetry in the functions. Separation of a function or of a partition. Solution of the problem of enumeration. Operation of upon the new functions. An example of enumeration 46
59—61. Correspondence with numbered diagrams 52


Chapter VI

The most general case of Distribution


62—74. Distribution when the boxes are identical. Multipartite numbers and their partitions. Distribution into similar boxes identified with the partitions of multipartite numbers. Solution of the problem by means of product-sums of certain combinations. Application of symbol . Simple particular cases 56
75—77. The most general case of distribution. Application to the distribution of identical objects. Elegant theorem of distribution which depends upon conjugate partitions. Some particular examples and verifications 65
78—81. Certain restricted distributions 67