Integer Partition

An integer partition is a way of writing a positive integer as a sum of positive integers, called its parts, without regard to the order in which the parts are listed, so that, for example, the number four has exactly five partitions, four itself, three plus one, two plus two, two plus one plus one, and one plus one plus one plus one. The number of partitions of a given integer, usually written as a function of that integer, grows very rapidly, and finding an exact or approximate formula for it was a long standing problem in number theory. The most celebrated result on the subject is the asymptotic formula found in 1918 by Srinivasa Ramanujan and Godfrey Harold Hardy, which gives an extremely accurate approximation for the number of partitions of large integers, and which was later refined by Hans Rademacher into an exact convergent series. Integer partitions appear throughout combinatorics, number theory and mathematical physics, including in the statistical mechanics of quantum particles.