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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 2 − 1 possibilities.Greek historian Plutarch discusses an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers.
The twelvefold way provides a unified framework for counting permutations, combinations and partitions.
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.
Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. Graph theory also enjoyed an explosion of interest at the same time, especially 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.
Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials.
Originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field.1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), 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, and would eventually become known as Pascal's triangle.Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.A mathematician who studies combinatorics is called a Basic combinatorial concepts and enumerative results appeared throughout the ancient world.Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas.