P n, 0 , which has no combinatorial meaning, is taken by convention as unity. A few points about these should be noted. This is a function of n and r, say f n, r. Equations, theorems, sections, examples, and problems are numbered consecutively in each chapter and are referred to by these numbers in other chapters. The emphasis is on methods of reasoning which can be employed later and on the introduction of necessary concepts and working tools.
These may be permuted in p! Combinations with Repetition The number sought is that of r-combinations of n distinct things, each of which may appear indefinitely often, that is, 0 to r times. If they do not, the number is P n — 1, r ; if they do, there are r positions in which the given thing may appear and P n — 1, r — 1 permutations of the n — 1 other things. Suppose the things are numbered 1 to n; then the combinations either contain 1 or they do not. This usage is followed here by taking the unqualified term, permutation, as always meaning n-permutation. The most natural method here seems to be that of recurrence and the rule of sum.
Rule of Product: If object A may be chosen in m ways, and thereafter B in n ways, both A and B may be chosen in this order in mn ways. The effectiveness of the enumerator in dealing with the numbers it enumerates is indicated in the examples following. In the language of statistics, U n, r is for sampling with replacement, as contrasted with P n, r which is for sampling without replacement. Using 2 , 1 may be rewritten and is also given the abbreviation The last is called the falling r-factorial of n; since the same notation is also used in the theory of the hypergeometric function to indicate n n + 1 · · · n + r — 1 , the word falling in the statement is necessary to avoid ambiguity. The number of combinations with repetition, 2 at a time, of 4 things numbered 1 to 4, by equation 10 , is 10; divided into two parts as in equation 9 , these combinations are The result in 10 is so simple as to invite proofs of equal simplicity. Now and if this is repeated until the appearance of f 1, 2 , which is unity, by the formula for the sum of an arithmetical series, or by the first of equations 8. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions.
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. Chapter 4 examines the enumeration of permutations in cyclic representation and Chapter 5 surveys the theory of distributions. These problems assume a certain amount of mathematical maturity. What are the generating functions and enumerators when the elements to. Another form is which implies the relation in rising factorials: namely, The result in 11 is only a beginning. An r- permutation of n things is an ordered selection or arrangement of r of them.
Equations, theorems, sections, examples, and problems are numbered consecutively in each chapter and are referred to by these numbers in other chapters. Notice that, in the first, the choices of A and B are mutually exclusive; that is, it is impossible to choose both in the same way. Distinct Things The first of the members of an r-permutation of n distinct things may be chosen in n ways, since the n are distinct. The number of those of the first kind is C n — 1, r — 1 , since fixing one element of a combination reduces both n and r by one; the number of those of the second kind, for a similar reason, is C n — 1, r. The author begins with the theory of permutation and combinations and their applications to generating functions. Chapter 3 contains an extended treatment of the principle of inclusion and exclusion which is indispensable to the enumeration of permutations with restricted position given in Chapters 7 and 8. An r- combination of n things is a selection of r of them without regard to order.
In subsequent chapters, he presents Bell polynomials; the principle of inclusion and exclusion; the enumeration of permutations in cyclic representation; the theory of distributions; partitions, compositions, trees and linear graphs; and the enumeration of restricted permutations. The number of arrangements on one shelf of n different books, for each of which there are m copies, is nm! The number of ways an r-hole tape as used in teletype and computing machinery can be punched is 2 r. The expression 1 + t n is called the enumerating generating function or, for brevity, simply the enumerator, of combinations of n distinct things. Hence, each place in an r-permutation may be filled in n different ways, because the stock of things is unaltered by any choice. Form the algebraic product Multiplied out and arranged in powers of t, this is or, in the notation of symmetric functions, where a1, a2, and a3 are the elementary symmetric functions of the three variables x1, x2, and x3. First, in either case, nothing is said of the features of the n things; they may be all of one kind, some of one kind, others of other kinds, or all unlike. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press.
Most of the proofs employ in one way or another either or both of the following rules. Each chapter includes a lengthy problem section, intended to develop the text and to aid the reader. The Number of Permutations of n Things, p of Which Are of One Kind, q of Another, etc. C n, 0 , the number of combinations of n things zero at a time, has no combinatorial meaning, but is given by 6 to be unity, which is the usual convention. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. Chapter 6 considers partitions, compositions, and the enumeration of trees and linear graphs.
Suppose the p like things are replaced by p new things, distinct from each other and from all other kinds of things being permuted. These rules are in the nature of definitions or tautologies and need to be understood rather than proved. For illustration, consider three objects labeled x1, x2, and x3. The set of numbers nl, n2, · · · , nk is called the specification of the things. Because it is so familiar, having been set forth for a generation in textbooks on elementary algebra, it is given here with a minimum of explanation and exemplification. It is also the number of arrangements of n unlike things into unlike cells or boxes, p in the first, q in the second, and so on, without regard to order in any cell, as will be shown in Chapter 5. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions.
The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. Chapter 3 contains an extended treatment of the principle of inclusion and exclusion which is indispensable to the enumeration of permutations with restricted position given in Chapters 7 and 8. The best of these, which may go back to Euler, is as follows. This book introduces combinatorial analysis to the beginning student. If they do not, there are f n — 1, r possible combinations.