Combinatorics is an area of mathematics concerned with counting, both as a means and an end in obtaining results, certain properties of finite structures. It is related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. To understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon. According to H. J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with the enumeration of specified structures, sometimes referred to as arrangements or configurations in a general sense, associated with finite systems, the existence of such structures that satisfy certain given criteria, the construction of these structures in many ways, optimization, finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other optimality criterion.
Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge in their objectives, their methods, the degree of coherence they have attained." One way to define combinatorics is to describe its subdivisions with their problems and techniques. This is the approach, used below. However, there are purely historical reasons for including or not including some topics under the combinatorics umbrella. Although concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite but discrete setting. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory and geometry, as well as in its many application areas. Many combinatorial questions have been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the twentieth century, however and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right.
One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist. Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In the 6th century BCE, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc. thus computing all 26 − 1 possibilities. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle. In the Middle Ages, combinatorics continued to be studied outside of the European civilization; the Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, these formulas may have been familiar to Indian mathematicians as early as the 6th century CE.
The philosopher and astronomer Rabbi Abraham ibn Ezra established the symmetry of binomial coefficients, while a closed formula was obtained by the talmudist and mathematician Levi ben Gerson, in 1321. The arithmetical triangle—a graphical diagram showing relationships among the binomial coefficients—was presented by mathematicians in treatises dating as far back as the 10th century, would become known as Pascal's triangle. In Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for algebraic combinatorics. Graph theory enjoyed an explosion of interest at the same time in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject.
In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics; the twelvefold way provides a unified framework for counting permutations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory
César Franck's Piano Quintet in F minor is a quintet for piano, 2 violins and cello. The work was composed in 1879 and has been described as one of Franck's chief achievements alongside his other late works such as Symphony in D minor, the Symphonic Variations, the String Quartet, the Violin Sonata; the work was premiered by the Marsick Quartet, with Camille Saint-Saëns playing the piano part, which Franck had written out for him with an appended note: "To my good friend Camille Saint-Saëns". A minor scandal ensued when at the piece's completion, Saint-Saëns walked off stage leaving the score open at the piano, a gesture, interpreted as mark of disdain; that manuscript is now in the Bibliothèque nationale de France The published form issued by Hamelle in 1880, carries the simpler dedication "À Camille Saint-Saëns". The work has been described as having a "torrid emotional power", Édouard Lalo characterized it as an "explosion". Other critics have been less positively impressed: Roger Scruton has written of the quintet's "unctuous narcissism".
There are three movements: Molto moderato quasi lento – Allegro Lento con molto sentimento Allegro non troppo ma con fuocoThe music has a cyclical character whereby a motto theme of two four-bar phrases, used 18 times in the first movement, recurs at strategic point in the work. Piano Quintet: Scores at the International Music Score Library Project
Cyberrebate.com, Inc. was an online retailer founded in May 1998 that went bankrupt in May 2001, after the collapse of the dot-com bubble. The company sold items at grossly inflated prices, as much as 10 times the list price, but promised customers a 100% rebate; the company relied on the assumption that 50% of its customers would neglect to apply for their rebate. Joel Granik, Joseph Lichter and Athan Vadiakas started the website on May 16, 1998. By November 2000, the company claims to have rebated $39 million to its customers. In January 2001, it was the third–ranked online retailer in the United States and had 7.7 million web users per month. The company filed for bankruptcy protection under Chapter 11, Title 11, United States Code on 16 May 2001, citing $83.3 million in liabilities and $24.5 million in assets. $80 million was due directly to customers in unpaid rebates. At the time of the bankruptcy filing and there were 9 customers that were due pending rebates of $79,000-$100,000 each. In April 2005, some creditors were awarded $0.08802 per dollar of allowed claims.
A second, final disbursement was made to creditors in August 2006 for $0.0006276 per dollar of allowed claims, or $1 for every $1,600 claimed