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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
Combinatorial designs date to antiquity, with the Lo Shu Square being an early magic square.One of the earliest datable application of combinatorial design is found in India in the book Brhat Samhita by Varahamihira, written around 587 AD, for the purpose of making perfumes using 4 substances selected from 16 different substances using a magic square.
3 out of 4638576 [1] or out of 580717, [2] if rotations and reflections are not counted as distinct, Hamiltonian cycles on a square grid graph 8х8. Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations).For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.
One of the earliest uses of analytic techniques for an enumeration problem came from Srinivasa Ramanujan and G. H. Hardy's work on integer partitions, [4] [5] starting in 1918, first using a Tauberian theorem and later the circle method.
Frontispiece of the book printed in 1690. The Dissertatio de arte combinatoria ("Dissertation on the Art of Combinations" or "On the Combinatorial Art") is an early work by Gottfried Leibniz published in 1666 in Leipzig. [1]
The order of battle across the four armies comprised 12 corps, containing a total of 48 divisions. Formed eight days after the Normandy landings, it initially controlled the First and the Third US Armies. Through various configurations in 1944 and 1945, the Twelfth US Army Group controlled the majority of American forces on the Western Front.
Alfonso XIII [a] (Spanish: Alfonso León Fernando María Jaime Isidro Pascual Antonio de Borbón y Habsburgo-Lorena; French: Alphonse Léon Ferdinand Marie Jacques Isidore Pascal Antoine de Bourbon; 17 May 1886 – 28 February 1941), also known as El Africano or the African for his Africanist views, was King of Spain from his birth until 14 April 1931, when the Second Spanish Republic was ...