If anything at all can be deduced from the two quotations at the top of this page, perhaps it is this: Combinatorics is an essential part of the human spirit; but it is a difficult subject for the abstract, axiomatising Bourbaki school of mathematics to comprehend. Nevertheless, the advent of computers and electronic communications have made it a more important subject than ever.

This is a textbook on combinatorics. It's based on my experience of more than twenty years of research and, more specifically, on teaching a course at Queen Mary and Westfield College, University of London, since 1986. The book presupposes some mathematical knowledge. The first part (Chapters 2–11) could be studied by a second-year British undergraduate; but I hope that more advanced students will find something interesting here too (especially in the Projects, which may be skipped without much loss by beginners). The second half (Chapters 12–20) is in a more condensed style, more suited to postgraduate students.

I am grateful to many colleagues, friends and students for all kinds of contributions, some of which are acknowledged in the text; and to Neill Cameron, for the illustration on p. 128.

I have not provided a table of dependencies between chapters. Everything is connected; but combinatorics is, by nature, broad rather than deep. The more important connections are indicated at the start of the chapters.

Combinatorics is special. Most mathematical topics which can be covered in a lecture course build towards a single, well-defined goal, such as Cauchy's Theorem or the Prime Number Theorem. Even if such a clear goal doesn't exist, there is a sharp focus (finite groups, perhaps, or non-parametric statistics). By contrast, combinatorics appears to be a collection of unrelated puzzles chosen at random.

Two factors contribute to this. First, combinatorics is broad rather than deep. Its tentacles stretch into virtually all corners of mathematics. Second, it is about techniques rather than results. As in a net, threads run through the entire construction, appearing unexpectedly far from where we last saw them. A treatment of combinatorics which neglects this is bound to give a superficial impression.

This feature makes the teacher's job harder. Reading, or lecturing, is inherently one-dimensional. If we follow one thread, we miss the essential interconnectedness of the subject.

I have attempted to meet this difficulty by various devices. Each chapter begins with a list of topics, techniques, and algorithms considered in the chapter, and cross-references to other chapters. Also, some of the material is set in smaller type and can be regarded as optional.

The complex and many-faceted problem of democracy will be dealt with in this paper only as it affects the situation at the present moment here in Germany. We shall go straight into the topic without further ado and without reflections of a general kind.

As is generally known, the present franchise for elections to the Reichstag was introduced by Bismarck for purely demagogic reasons in his famous ultimatum to the Frankfurt Federal Diet when he championed this principle in the face of grave reservations from the liberals of the time. His motives had to do partly with foreign policy objectives, and partly with the domestic political aim of realising his Caesarist ambitions in defiance of the (at that time) recalcitrant middle classes. Admittedly, his hopes that the masses would respond conservatively were disappointed, but the splitting of precisely those social strata which are so characteristic of the structure of modern society into two classes existing in close proximity and hence in hostility to one another (the bourgeoisie and the proletariat) later made it possible (as Prince Hohenlohe observed) to exploit the cowardice (Hohenlohe called it ‘timidity’) of the bourgeoisie in the face of ‘democracy’ and thereby preserve the rule of the bureaucracy. This cowardice is still having its effects today. The fact that it was perfectly possible to be a democrat and yet to reject Lassalle's enthusiasm for that form of franchise under the circumstances of the time, is evident, for example, from the position taken by Eduard Bernstein in his introduction to Lassalle's writings.

May I be permitted to add to the above account, which has kindly been made available to us, some remarks about the political current in which the draft originated. The question of the extent to which the draft might assume practical importance in forthcoming political discussions is one we shall leave aside. For our purposes it suffices that the draft is symptomatic of a particular mode of political thought amongst outstandingly able and idealistic Russian patriots for whom we have complete personal sympathy, regardless of any success, given the enormous difficulties in their situation, their work ultimately has. The fact that they are generally no friends of German culture – indeed often its bitter enemies on Russian soil – and the fact that they are predominantly hostile to Germany in political matters does nothing to change my attitude.

The draft has been worked out by members of the ‘Union of Liberation’ (Soyuz Osvobozhdeniya) and is formally one of the projects debated at the congresses of the members of the zemstvos and the Duma. Let me say a few words about both organisations which are the bearers of the liberal and democratic movement. Although its official constitution did not take place until January 1904 in Petersburg, the ‘Union of Liberation’ was founded in the summer of 1903 during an ostensible group holiday in the Black Forest under the chairmanship of the estate-owner Petrunkevich who, along with the zemstvo of Tver, had been disciplined by Plehve.

The discussion (in Die Frau) about the meaning and purpose (Sinn) of our war could perhaps be augmented by placing more emphasis on a point, the importance of which you in particular will readily appreciate, namely our responsibility before history – I can only put it in these rather pathetic terms. The facts themselves are plain enough.

Any numerically ‘large’ nation organised as a Machtstaat finds that, thanks to these very characteristics, it is confronted by tasks of a quite different order from those devolving on other nations such as the Swiss, the Danes, the Dutch or the Norwegians. There is of course a world of difference between this assertion and the view that a people which is ‘small’ in numbers and in terms of power is thereby less ‘valuable’ or less ‘important’ before the forum of history. It is simply that such nations, by their very nature, have different obligations and therefore other cultural possibilities. You are familiar with Jakob Burckhardt's arguments, which have caused so much astonishment, about the diabolical nature of power. In fact this evaluation is a wholly consistent one, when considered from the standpoint of those cultural values which have been entrusted to a people, such as the Swiss, who are not able to bear the armour of great military states and who therefore have no historical obligation so to do.

Karl Emil Maximilian Weber was born in Erfurt in 1864. His father, Max Weber Sr, was a lawyer and a deputy in the Prussian Chamber of Deputies for the National Liberal Party from 1868 to 1882 and from 1884 to 1897. He was also a member of the Reichstag from 1872 until 1884. Weber's mother, Helene Fallenstein Weber, had an interest in questions of religion and social reform which she did not share with her husband.

The Weber household in Berlin attracted a large number of academics and politicians, including von Bennigsen, Dilthey, Theodor Mommsen and Treitschke. The discussions which took place there must have made a strong impression on the young Weber. In 1882 Weber began his studies at Heidelberg University. His main subject was law but he also attended courses in political economy, history, philosophy and theology. He moved to Strasbourg in 1883 where he combined his year of national service with study at the university. In 1884 Weber continued his studies in Berlin. Here he attended courses in law, including Gierke's course on German legal history. Weber was not impressed by the lectures of Treitschke which, because of their extreme nationalism, he considered to be little more than demagogy and propaganda. After graduation Weber did not find the practice of law sufficiently stimulating and continued his studies in the field of political science (Staatsmissenschaft) as well as in legal and economic history.

Preface

I was prompted to publish the following arguments by the opposition rather than the assent which they elicited from my audience. They offer colleagues in the same discipline, and others, new information only on points of detail, and the occasion that gave rise to them explains the special sense in which alone they lay claim to the name of ‘science’. Essentially, an inaugural lecture is an opportunity to present and justify openly the personal and, in this sense, ‘subjective’ standpoint from which one judges economic phenomena. The exposition on pages 17–20 was omitted for reasons of time and in view of the audience, while other parts of the argument may have assumed a different form when I was actually delivering them. It should be noted that the opening remarks give a very simplified account of events which were naturally a good deal more complicated in reality. During the period 1871–85 the population movements in individual districts and communities in West Prussia were not uniform, although they changed in characteristic ways, and they are much less transparent than the examples selected here. In other instances the tendency I have tried to illustrate from these examples is subject to the influence of other factors.

The first president of the Reich was elected by the National Assembly. In future the president of the Reich absolutely must be elected directly by the people. The decisive reasons for this are as follows:

(1) Regardless of whatever name it is given and whatever changes are made to its powers, the Bundesrat will under all circumstances be carried over into the new constitution of the Reich in one form or another, for it is utterly Utopian to imagine that the bearers of governmental authority and state power, namely the governments installed by the peoples of the individual free states, will allow themselves to be excluded from the process of shaping the will of the Reich and above all from the administration of the Reich. It is therefore essential for us to create a head of state resting unquestionably on the will of the whole people, without the intervention of intermediaries. Indirect elections have been abolished everywhere; are they then to be preserved here, for the election of the highest office? That would be regarded, quite rightly, as a mockery of the democratic principle in favour of the interest members of parliament have in horse-trading, and it would discredit the unity of the Reich.

(2) Only a president of the Reich who has the votes of millions of people behind him can have the authority to initiate the process of socialisation.