# Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/13

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