Research On The History Of Combinatorics

Combinatorics,. formed as a branch of mathematics in 1960s, has a long history. It holds a rapid speed of development in recent years. Its history will be stated in six parts as following:1. It's developed by three aspects that the origin of combinatorial idea is traced to the ancient east. The order 3 magic square appeared in ancient China is the oldest instance for combinatorial design. There also were a large number of examples on magic square in ancient India and Arab. Some mathematical games and recreations acted as an essential role in the development of earlier combinatorics.2. This part gives a study on the some objects in combinatorics researched in the middle ages. Those researches are substantially composed by search for the formulae of permutation and combination, the arithmetical triangle and the magic square. Both the east and west gave a contribution to those subjects.3. Interpret the contents both on the pile and difference in Precious Mirror of the Four Elements (Si Yuan Yu Jian) and the arithmetical triangle in Treat on the Arithmetical triangle (Traite du Triangle Arithmetique). Both works are the much earlier books study the combinatorial identities in system. Furthermore, the modern researches on the identities are investigated which are derived from the binomial coefficients, inversion relations and partition polynomials.4. The history of primary problems in enumerative combinatorics is explored in some special subjects. (1) Discuss earlier researches on such counting functions, as Fibonacci numbers, Catalan numbers and Stirling numbers. (2) Interpret in detail the contributions that the west and east countries had given to the subject-sum of powers, especially P. Fermat, B. Pascal and J. Faulhaber's methods to this subject. (3) Expound the outline of the development of partition of integer. (4) Analyze the evolution of problem of derangement and the principle of inclusion-exclusion. (5) Relate the arisen history of P61ya theorem and explore the method that P6 -lya gave this theorem.5. The evolution of four fundamental components in design theory is delivered. (1) State the appearance of block design and it's resolution. (2) Formulate the history of OLS (orthogonal Latin squares) and show the role that Euler's conjecture and MacNeish's conjecture on OLS played in the progress of study on Latin square. (3) State briefly the motivation that finite projective plan and finite field offer to the development of combinatorial design.6. The history of Ramsey theory and SDR in combinatorial set theory is briefly discussed. In short, I hope to present a concise history of Combinatorics through above discussion. Sothe development of this mathematical branch can be distinctly acquainted.