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

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 $D_{m}$	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 $D_{m}$	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 $D_{m}$ 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 $D_{m}$. 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