# Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/14

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Contents

viii

Chapter IV

Distribution when Objects and Boxes are equal in number

 art. page 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