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

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

Pages   (key to Page Status)

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 ${\displaystyle D_{m}}$ 22 30—31. Operation of ${\displaystyle D_{m}}$ upon a product of functions. Connexion with the compositions of ${\displaystyle m}$ 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 ${\displaystyle D_{m}}$ 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 ${\displaystyle D_{m}}$ 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 ${\displaystyle D_{m}}$ 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 ${\displaystyle D_{m}}$ 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 ${\displaystyle D_{m}}$ 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 ${\displaystyle D_{m}}$. 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