# An Introduction to Combinatorial Analysis

JOHN RIORDAN
Pages: 256
https://www.jstor.org/stable/j.ctt7zv5hs

1. Front Matter
(pp. i-vi)
2. Preface
(pp. vii-x)
John Riordan
(pp. xi-xii)
4. CHAPTER 1 Permutations and Combinations
(pp. 1-18)

This chapter summarizes the simplest and most widely used material of the theory of combinations. 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. The emphasis is on methods of reasoning which can be employed later and on the introduction of necessary concepts and working tools. Among the concepts is the generating function, the introduction of which leads to consideration of both permutations and combinations in great generality, a fact which seems insufficiently known.

Most of the proofs employ in one...

5. CHAPTER 2 Generating Functions
(pp. 19-49)

It is clear from the discussion in Chapter 1 that a generating function of some form may be an important means of unifying the treatment of combinatorial problems. This could have been predicted from the De Morgan definition of combinatorial analysis cited in the preface, for if the latter is a means of finding coefficients in complicated developments of given functions, then these functions may be regarded as generating the coefficients, and their study is the natural complement to the study of the coefficients. The beginnings of this study are examined in this chapter.

First, the informal discussion of Chapter...

6. CHAPTER 3 The Principle of Inclusion and Exclusion
(pp. 50-65)

This chapter is devoted to an important combinatorial tool, the principle of inclusion and exclusion, also known variously as the symbolic method, principle of cross classification, sieve method (the significance of these terms will become apparent later). The logical identity on which it rests is very old; Dickson’sHistory of the Theory of Numbers(vol. I, p. 119) mentions its appearance in a work by Daniel da Silva in 1854, but Mont-mort’s solution in 1713 of a famous problem, known generally by its French name, “le problème des rencontres” (the number of permutations ofnelements such that no element...

7. CHAPTER 4 The Cycles of Permutations
(pp. 66-89)

Permutations may be regarded in two ways: (i) as ordered arrangements of given objects, as in Chapter 1, and (ii) as derangements of a standard order, usually taken as the natural (alphabetical or numerical) order, as in this chapter. These two ways are related as noun to verb, or as object to operator; the second is that naturally used in the study of permutations in the theory of groups.

To indicate completely a permutation of numbered elements as an operator, a notation like

$\downarrow \left( \begin{array}{l}12345 \\25431 \\\end{array} \right)$

is required, with the arrow indicating the direction of the operation, the first line the operand,...

8. CHAPTER 5 Distributions: Occupancy
(pp. 90-106)

Distribution has been defined by MacMahon, 1, as the separation of a series of elements into a series of classes; more concretely, it may be described as the assignment of objects to boxes or cells. The objects may be of any number and kind and the cells may be specified in kind, capacity, and number independently. Order of objects in a cell may or may not be important. When the number of assignments is in question, the problem is said to be one of distribution; when the number of objects in given or arbitrary cells is in question, the problem...

9. CHAPTER 6 Partitions, Compositions, Trees, and Networks
(pp. 107-162)

According to L. E. Dickson, 4, to whose account of the history the reader is referred for many interesting results, partitions first appeared in a letter from Leibniz to Johann Bernoulli (1669). The real development starts, like so much else in combinatoric, with Euler (1674),

The use of partitions in specifying a collection of objects of various kinds has already appeared in Chapter 1, and naturally calls for an enumeration. A partition by definition is a collection of integers (with given sum) without regard to order. It is natural, therefore, to consider along with partitions the corresponding ordered collections, which,...

10. CHAPTER 7 Permutations with Restricted Position I
(pp. 163-194)

This chapter is devoted to the enumeration of permutations satisfying prescribed sets of restrictions on the positions of the elements permuted. The permutations appearing in the problème des rencontres, described and solved in Chapter 3, provide the simplest example of a restricted position problem in which each element has some restriction (elementiis forbidden positioni). A related problem is that called by E. Lucas, 11, the “problème des ménages”; this asks for the number of ways of seatingnmarried couples at a circular table, men and women in alternate positions, so that no wife is next to...

11. CHAPTER 8 Permutations with Restricted Position II
(pp. 195-238)

In this chapter the discussion of the topics introduced in Chapter 7 is continued, moving on to the staircase chessboards, the most famous example of which is the ménage problem, to Latin rectangles which are closely associated, and, finally, to the trapezoidal and triangular boards which appear in Simon Newcomb’s problem. An interesting use of the last is in the recreation known as the problem of the bishops. Each of these topics has a growing end and the treatment of the text and its continuation in the problems merely serve to define the open regions; a striking example is that...

12. Index
(pp. 239-244)